# Children's Guild Part 1

( D.C. Public Charter School Board Charter Petition for The Children's Guild District of Columbia Public Charter School ( March 3, 2014 ( ( Table of Contents Executive Summary........................................................................................................................ 1 A. Educational Plan .............................................................................................................. 5 1. Mission and Purposes of the Proposed Public Charter School .............................................. ..... 5 2. Goals ....................................................................................................................................... 16 3. Charter School Curriculum .......................................... .......................................... ................... 24 4. Support for Learning ....... ...... ................................................................................................... 54 B. Business Plan ................................................................................................................. 77 1. Planning and Establishment ............................................................................. .. ... .................. 77 2. Governance and Management .... ....................... ..................................... ................................ 81 3. Finance ......................................... ..................................................... ... .. ................................ 106 4. Facilities ............................................. .. ...................................................... .. .......................... 112 5. Recruiting and Marketing .. ........................................... .............................. ........................... 114 C. Plan of Operation .......................................................................................... ............... 116 1. Student Policies and Procedures ............................................................................................ 116 2. Human Resource Information ........................ .. ............ .................................................. ........ 122 3. Implementation of the Charter ....................................... .. ..................................................... 131 ( Appendices D. Certifications E. Budget F. Resumes, Board Member Agreements, and Statements of Interest and Qualifications G. Conflict of Interest H. Demographic Analysis I. Required Documents J. Curriculum K. Parent and Student Satisfaction Surveys L. Culture Card M. Behavior Motivation and Intervention System N. Assessment Habits 0. Student Behavior Management Process P. PBIS Matrix Q. Discipline Policy R. Letters of Support S. Grievance Policy T. Bylaws U. Conflict of Interest Policy V. Audited financials for Monarch Academy Baltimore and Glen Burnie W. Accounting Procedures Manual XYZ. Recruitment & Interview Policy Personnel Handbook ( The Children's Guild, Ltd., DC Public Charter School Page 2 Applicant Information Sheet ( New Charter School Request for Approval This application is a request to establish and operate a Public Charter School as provided in the District of Columbia School Reform Act of 1995, as amended. Name of Proposed Charter School: The Children's Guild District of Columbia Public Charter School Name of Entity Applying for Charter Status in D.C.: The Children's Guild. Ltd. Contact Person: Andrew L. Ross Ph.D. Address: 6802 McClean Boulevard. Baltimore. MD 21234 Daytime Telephone: 410-444-3800 x1157 Fax: 410-444-4695 Email: ross@childrensguild.org Name of Person Authorized to Negotiate: Andrew L. Ross. Ph.D. (until board is appointed. then board chair {Must be memberoflocalfounding group and notserving as a consultantorafjiliated with an educational service provider.) Authorized Signature: ~~~ Proposed Year One Budget: $ 11.535.098 Proposed Start Date: August. 2015 ( Start-up Information Year One 5/Kindergarten 14/81h Grade 450 Starting Age/Grade 5/Kindergarten Highest Age/Grade 14/81h Grade Total Number of Students/ Enrollment Ceiling 450 Two 5/Kinderga rten 14/81h Grade 450 Capacity Proposed Location of School (address or area of city): 5600 E. Capitol. NE. Washington. DC 20019 Name of Educational Service Provider (if applicable): __,Tc.:.h=e:....::C=h....,.i....,ld=r=e,....n_,'s_,G"->u=i....,ld,___ _ _ _ _ _ __ Type of Application (Check One) 0 Conversion of Existing Public School ( 0 Conversion of Existing Private School 0 No 1!4New School If conversion, name the school being c o n v e r t e d : - - - - - - - - - - - - - - - - - - If conversion, do you wish to retain the existing school site? 0 Yes LEA Status: Will the school elect to be treated as a Local Education Agency (LEA) for purposes of Part B of the IDEA and Section 504 of the Rehabilitation Act of 1973? 1!4 Yes 0 No 15 ( Executive Summary Our Charter History The Children's Guild has a 60 year history of successfully educating students with special needs. Using an innovative philosophy known as Transformation Education, The Children's Guild developed and refined its model to eventually produce both nonpublic schools and residential care programs whose outcomes substantially bettered the national averages for those services and eclipsed the outcomes of more prominent providers in the region and nation. In 2009, the Anne Arundel County (MD) schoo l district invited The Ch ildren's Guild to extend Transformation Education to the charter school world. In 2010, The Chi ldren's Guild opened its first charter school, Monarch Academy Glen Burnie in Anne Aru ndel County. Transformation Education Expanded Transformation Education transferred effectively to the charter schoo l arena. Monarch Academy Glen Burnie (MAGB) has become well-known and well-respected in the county. Today, MAGB has more students on its waiting list than it has tota l seats available in its filledto-capacity building. Anne Arundel County has sought and approved charters for two add itional Monarch Academies, Monarch Global Academy to open in August 2014 and Monarch Central Academy to open in August 2015. In addition, Baltimore City Public Schools approved their own Monarch charter schoo l in 2011 and Monarch Academy Baltimore City now resides in the beautifu lly renovated Coca Co la bottling plant on Kirk Avenue on the city's east side. As Transformation Education was successfully being introduced into the charter school setting, The Children's Guild's nonpublic schoo ls in Baltimore and Chi llum, Maryland continued to exceed the national averages for return-to-public-school rate. George Washington University studied the Transformation Education approach and found it to be more successful than comparable models. Our group home programs were so successful a book was written about them and Transformation Education even expanded to treatment foster care and publ ic school therapist placements through our outpatient mental health center. Transformation Education is the Key How does an organ ization successfully operate high impact programs in education, residential care, foster care, tra ining, consulting and community-based therapy? The Children's Guild's philosophy, Transformation Education, is the key factor. Transformation Education believes that culture is the most powerfu l teaching tool known to human beings. As a resu lt, we bu ild our schools, group homes, and other programs by first creating a culture that will communicatepowerful ly and consistently-the values of caring, contribution and commitment and the skills of vision, courage and will to every person coming in contact with that culture. We then monitor, measure and maintain that culture so that its power to produce growth and change in children, youth and adults is refined and strengthened. ( ( The Children's Guild, Ltd., DC Publ ic Charter School Page 3 ( Assembling Our Best In addition to implementing our cultural model, The Children's Guild will bring it s best educational and behavioral practices from all of its programs to create the special hybrid of The Children' s Guild District of Columbia Public Charter School (CGDC). While literally dozens of proven strategies, interventions and assessments will be utilized, five best practices form the foundation of our proposed school over the bedrock of Transformation Education. These practices include the TEACCH structures developed at the University of North Carolina; ProjectBased Learning strategies and tools; a multi-dimensional emphasis on Character Development; a commitment to Arts Integration as a support to high rigor and; a state-of-the-art Diagnostic and Evaluation center that uses highly-skilled professionals and practices to identify the programs and services each student needs to reach his or her optimal academic potential. This assembly of "best practices" layered over the Transformation Education philosophy w ill provide the District, its students and other charters with a school to partner and collaborate with to serve the diverse special education needs of the com munity. A New Kind of Charter CGDC will look like no other school in the District, charter or traditional public. It will seek special needs students while providing general education students and families with many attractive features and opportunities. It will not seek to compete with but to comp lement the existing charters in the District and partner with its fellow charters to help each and every student get the right services leading to the best possible educational result. Utilizing the special qualifications of The Children's Guild, CGDC will introduce a " new kind of charter" and f ill a growing need in the District's charter school continuum . CGDC will be a schoo l equipped to serve any and all students enrolled-at both ends of the educationa l spectrum and every level in between. CGDC will also help other charters better se rve all of their students and thereby contribute to the consistency and stabi lity of charter school enrollments throughout the District. A Special Education Specialist for DC Charters Now, The Children's Guild seeks to bring Transformation Education and its long history of successfully serving special needs students to the District of Columbia Charter School network through this application to open The Children's Guild District of Columbia Public Charter School in Ward 7. CGDC would fi ll the District's charter school network's need for a "special education specialist" in the charte r continuum that can provi de technical assistance, training, consu ltation and diagnostic services for other charters as well as effectively serve its own student population. The unique feature of CGDC is that it would assertively seek a specia l education population of 60% of its student body. Because of its long history and demonstrated success in both the nonpublic and the charter arenas, The Chi ldren's Guild is uniquely qualified to operate a majority special education charter school and also provide challenge and rigor to general population students at the same time. ( ( The Children's Guild, Ltd., DC Public Charter School Page 4 ( education choices. In addition, creating openings for new students in older grades recognizes that students who are struggling in other settings may need the option to transfer into a school more suited for their needs. Many of the existing K-8 schools take mainly returning students and therefore have few vacancies. Serving proportionally older students will help us fulfill our mission of servin g students with special needs, since older students are more likely to be identified for special education than kindergartners . CGDC will also provide ESOL services for non English language learners in addition to a continuum of special education services : Levell : The general education classroom, with supplementary aids and services such as special education teacher su pport (inside of general education setting); Levels 2 and 3: A special education cla ssroom, for part of the school day, I with the student spending the remainder of the day in the general education classroom or in activities w ith students who do not have disabilities (combinat ion inside and outside of general education setting); Level 4: A self-contained special education classroom full-time. The D.C. 2013 Equity Report indicates that 13% of the District of Columbia's total school population is special education . CGDC is proposing an overall population of 450 students of which its goal is that 60%, or 270 will be students that need special education services. CGDC will proactively develop an outreach and recruitment strategy in order to reach that number, and will have appropriate operat ional plans in place in case it does not reach these leve ls, especially in its first year. CGDC anticipates that these students will be distributed among the different level s of special education based on the rate of these levels in the city's special education population as a whole, as reflected in the equity report. CGDC also projects that the schoo l will enroll approximately 6% of students with Limited English Proficiency, based on the DC average as published in the equity report. In accordance with the D.C. Equity Report, the special education distribution would be anticipated as follows: Proposed Percentage of student population Level 1 Level 2 Level3 Level4 Limited English 36% of 270 32% of 270 12% of 270 20% of 270 6% of 450 total students) Proposed number of students in each subgroup 98 86 32 54 27 ( The proposed enrollment matrix includes enrollment of students in all grades in the opening year, 2015-2016. The average class size will be 22 students with a teacher and an aide for each The Children's Guild, Ltd ., DC Public Charter School Page 6 ( kindergarten class. Grades 1 - 8 will have a shared aide for each grade level. Our charter schools in Anne Arundel County and Baltimore City have a similar student-teacher ratio and have experienced success with this model. We anticipate with the student population in Ward 7 that this ratio will provide the support needed for every student to access learning and achieve to their capacity. Please note that our experience is that not all grade levels enroll an identical number of students and that we will adjust the number of classes as necessary to fit CGDC's actua l enrollment. Anticipated Student and Classroom Distribution Projected Population 450 Grades served Average number of students per grade K-8 Grades K-3 has 22 4th grad e - 44 Grades 5-8 has 66 Average number of students per class Tota l numbe r of enrolled students Grade Vl +-' 22 450 Year 2 Year 3 '+- Year 1 '+Vl ( k c: 0 0 aJ ..... "C ..... ::l l/) E .c ..... 0 ..... aJ 0 0 E '+- 0 '+Vl +-' Vl E ..... c: 0 Vl ..... .c c: aJ aJ ..... c.. E ~ ::l ro z-u E~ ::l ::l z t; aJ "C ::l l/) 0 ..... ..... aJ .c ..... 0 ..... aJ 0 0 E 0 '+Vl +-' Vl +-' Vl ..... .c c: aJ +-' c.. E ~ ::l ro z-u E ~ ::l ::l z t; aJ "C ::l l/) c: E 0 0 ..... 0 .c ..... 0 ..... aJ E 0 '+- 0 ..... .c c: aJ Vl +-' ..... +-' c.. aJ E ~ ::l ro z-u E~ ::l ::l z t; 1 2 3 4 5 6 7 8 Level 22 22 22 22 22 22 22 22 22 9 1 1 1 1 2 3 3 3 3 22 22 22 22 44 66 66 66 66 22 22 22 22 22 22 22 22 22 9 1 1 1 1 2 3 3 3 3 22 22 22 22 44 66 66 66 66 22 22 22 22 22 22 22 22 22 9 1 1 1 1 2 3 3 3 3 22 22 22 22 44 66 66 66 66 7 54 7 54 7 54 4 Total 25 450 25 450 25 450 ( The open ing year of CGDC we are anticipating one classroom in grades kindergarten through 3rd grade, 2 classrooms in grade 4, and 3 classrooms in grade 5-8. In add it ion to these genera l education classrooms, we will have self contained special education classrooms to accommodate level 4 students with 9 or fewer students per classroom . The Children's Guild, Ltd., DC Publ ic Charter School Page 7 ( The use of the Transformation Education (TranZed) philosophy is a key feature of CGDC. The foundational beliefs and values of TranZed are outlined below: Foundational Beliefs • Life is a journey of personal growth ascending from a focus on self to a focus on family, community and world. Education results in personal growth with the ultimate goal of using that growth to contribute to a cause larger than oneself. • Culture is the most powerfu l force available to a school for transmitting pro-socia l va lues and transforming one's mindset. How one learns to think powerfu lly impacts life success. CGDC will teach students to think globally, problem-solve creatively and apply critical thinking skills as they engage in their life journey. • The school culture is a critical tool for transmitting values and fostering emotional and intellectual growth. • School leaders' management efforts focus on aligning faculty and student be liefs with the foundational beliefs and va lues of the school's mission. Core Values In addition to its foundational beliefs, TranZed's core values are caring, contribution and comm itment. To care is to be awa re and to be engaged both intellectually and emotionally. Cont ribution is caring in action and commitment is contribution over t ime. TranZed's Character Development efforts are focused on developing these core values in both students and staff. TranZed is the overarching organizational phi losophy found in all Children's Guild programs and services. The foundational theory base of TranZed integrates the fields of anthropology and neuroscience. The Children's Guild has developed a repertoire of specific tools and systems needed to make the research on learning from anthropology and neuroscience accessible and understandable to the teachers and administrators serving children . More about the Transformation Education philosophy and the successful outcomes achieved through its app lication to a variety of child-serving settings can be found in : Ross, A., Grenier, G. and Kros, F., Creating the Upside Down Organization: Transforming Staff to Save Troubled Children (2005, Children's Gui ld Press, Baltimore, MD.). Transformation Education recognizes culture as a child's f irst and most effective teacher. Therefore, TranZed focuses on aligning the faculty's mindsets, the school's systems, its physical environment, and its instructional approach to create a culture that is consistent with the school's beliefs, mission, and values. It also helps the school remain innovative and flexible enough to meet the individual needs of the children it serves while successfully navigating the sociat political and economic environment in which it operates. Culture is defined by anthropologist Charles Case as follows : ( ( In a very fundamental sense, culture is the most human part of man's existence. It encompasses those aspects of being that are learned, those regularities that are acquired, as The Children's Guild, Ltd., DC Pub lic Charter School Page 8 ( things that are gained through association with other humans. It is the social heritage that has developed out of the biological responses in the life process. It is the web of relationships holding people together in various viable groups. It is the structure of predictability in the behavior of the members of society, which tells each person who he is and who other people are. It provides the techniques for dealing with life's problems, and for directing the shape of one's existence . . .. culture . .. is a guiding system, a behavioral map, a grammar of behavior that leads one to places unsuspected, by paths unknown and perhaps even against one's will. It is constantly present working to shape behavior in its outward form . [Case, C., Culture, The Human Plan {1977} pp. 16-17}. TranZed understands that every aspect of the school's culture (the teachers, the policies, the environment, the curriculum, etc.} radiates messages to students about the beliefs, values and desired behaviors in the schoo l environment. TranZed culture is an intentional design that rad iates the school's beliefs, mission and values . TranZed integrates the current research from the neurosciences into its approach to instruction and school management. Brain-compatible education utilizes knowledge about how the brain learns naturally and is based on what is currently known about the actual structure and function of the human brain at varying developmental stages. Using the latest neural research, educational techniques that are brain-friend ly provide a biologically driven framework for creating effective instruction, management and the physical design of the school. Transformation Education is committed to t he intentional selection and implementation of best practices inspired by brain research. Children have powerful neural networks that reflect their adaptation to their own culture . Since the chief concern of childhood is physical and emotional survival, children are particularly attentive to cu ltural messages and construct strong neural networks as they learn to navigate the values, expectations and behaviors modeled in the cu lture. By immersing children in a prosocial environment, the brain's remarkable capacity to learn and adapt can result in substantial, long-lasting and positive change. TranZed also recogn izes that all human beings have a basic need to express their thoughts and feelings. Transformation Education recognizes the power of the artistic expression in supporting the neurobiological development of the brain in ways that enhance the pro-socia l and academic performance of students. Research shows that embedding creative expression and movement within academic learning allows the brain to make complex perceptual maps and provides an increased likelihood of engaging student emotions. In addition to the overarching Transformation Education philosophy, the primary components of the CGDC schoo l model include a number of key methods of instruction . CGDC will employ several identified best practices to serve as its foundation for building skills in the areas of academics, creative problem solving, and critical thinking and to support Character Development and a commitment to service. The primary components of CGDC include the TEACCH model, Project Based Learning, Character Development and Arts Integration. The Children's Guild, Ltd., DC Public Charter School Page 9 ( ( ( These key methods of instruction support both the mission and philosophy of CGDC in order to meet the learning needs of special education students, Engl ish language learners, and students who need to be provided with a challenging curriculum . The TEACCH approach provides teachers with the essential instructional structures that allow every student to succeed despite the diversity of learning styles they bring to the classroom . Project Based Learning enhances student engagement and relevance, fueling the motivation students need to develop and maintain to succeed in an academically rigorous environment. Character Development teaches both thinking and specific behaviors needed for success in life. Arts Integration throughout the curriculum supports academic rigor and provides every student opportunities for expressing themselves and interpreting their world . b. Educational Needs of the Target Student Population Educational Needs and Demograph ic Analysis CGDC will serve a citywide population but as it expects to be located in Ward 7 it believes that the ethnic demographic makeup will mirror that of the community in which the school is located. The chart below outlines the demographics of students enrolled in DCPS schools in Ward 7: ( 2013-2014 Student demographics for students in Ward 7 include: 97% Black 3% Hispanic/Latina <1% White <1% English Language Learners 99% Free and Reduced Meals (FARMS) 20% Special Education DCPCSB Ward 7 Student Demographics Nat A mer <1% Spec Pacif <1% <1% White 8% FARMS 66% 100% 73% 3% 81% 82% 2% <1% 84% 100% 3% 82% 1% 2% 3% LEP 9% Ed 13% 13% 13% 13% 14% 11% 32% 7% School DCPCSTotal Friendship Junior Academy SEED PC KIPP DC Key Cesar Chavez PCMS KIPP DC Promise Maya Angelou LowerMS DC Prep Grades Pre-K-12 4-8 6-1 2 5-8 6-8 1-4 7-8 4 Enrollment 80230 683 341 327 409 415 196 431 Asian 1% Black 73% 99% 99% 96% 85% 96% Hispanic 16% 1% 1% <1% 15% 1% <1% 2% Multi 1% <1% 99% 95% The Children's Guild, Ltd., DC Public Charter School Page 10 ( Although the 2013 District of Columbia School Equity Report indicates a special education enrollment of 13% for the 2012-2013 school year in the District, a higher percentage of special education enrollment is expected at CGDC. Due to The Ch ildren's Guild history of 60 years in special education, the enrollment at our Monarch Academy Glen Burnie Campu s (MAGB) has been at or above the district average in special education. During MAGB's second year, special education enrollment increased over its first year of existence so we expect CGDC to mirror the trend we are experiencing at our two existing charter schools. CGDC recognizes that our expectation of enrolling 60% special education population is not aligned with the current special education population enrollment in Ward 7. Therefore, CGDC expects to market throughout the District to recruit referrals of special education services. However, in the event that CGDC recruits less than 60% special education students (i.e., 20 to 30%), we anticipate th is percentage to increase over time based on past experience, referrals and recruitment. To reiterate, CGDC will promote its interest, experience and capability in serving children with special education needs from across the DC metro area through marketing strategies in hope of attracting a student body population of 60% with special needs. CGDC expects such a student population to need targeted interventions; rigorous learning expectations with structures and processes supporting student progress; and high engagement with opportunities for student leadership that builds ownership of learning and drives ach ievement. As indicated below, the District's student performance on the NAEP of 2011 is significantly below the national average in reading and math for both the general population and disabled students. 45 40 35 30 25 20 15 10 5 0 4th 8th 4th Grade Disabled 8th Grade Disabled ( • DC Reading • National Average Reading • DC Math • National Average Math In mathematics, 5% of 4th grade students with disabilities scored proficient and above as compared to the national average of 17% and 2% of 8th grade students with disabil ities scored proficient and above as compared to national average of 9%. Data from the 2012-2013 DCAS assessment yields the following resu lts from DCPS schools and DC PCSB (general education) in Ward 7: ( The Chi ldren's Guild, Ltd., DC Public Charter School Page 11 ( DCAS 2012-2013 Math Reading DCPS Ward 7 DCAS Results Below Basic 25% 4% Basic 39% 46% Proficient 25% 30% Advanced 5% 2% DC PCSB Ward 7 DCAS Results School DC PCS Total grades Pre- K12 enrollment 80230 Subject Math Read Friendship Junior Academy "4-8 683 Math Read SEED PC "6-12 341 Mat h Read KIPP DC Key "5-8 327 Mat h Read All 53 % 49 % 45 % 31 % 68 % 47 % 86 % 76 Cesar Chavez PCMS % 42 % 48 % 63 "1-4 415 Math Read Maya Angelou Lower MS "7-8 196 Math Read DC Prep "4 431 Math Read % 56 % 38 % 36 % 59 % 55 % 37% 36% 58% 55% 38% 36% 58% 54% 11% 2% 56% 57% 63% 62% 76% 37% 45% 74% 44% 51% 13% 86% 86% 32% 47% SO% Asian 86% 74% Black 47% 44% 45% 31% 68% Hispanic 59% 52% Multi 83% 80% Pacif 67% 52% White 91% 92% FARMS 46% 42% 45% 31% 66% LEP 50% 40% Spec Ed 24% 19% 12% 9% ( "6-8 409 Math Read KIPP DC Promise These result s illustrate that the majority of Ward 7 students are performing below the proficient level in both read ing and math. The highest percentage of students performed at the basic level. In order to reduce th is be low level achievement, CGDC will provide rigorous differentiated instruction that meets the needs of students performing below grade level expectancy. As with many urban environments, CGDC expects its student body to need support in selecting and ana lyzing behaviors that are consistent with rigorous learning and with becoming responsible citizens in the school's learning community, th e communities where students live and the school resides, and in the global context of t oday's interconnected world . The Children's Guild possesses 60 years of experience in operating schools th at serve the needs of chi ldren and youth with emotional disabilities, learning disabled students, and children wit h The Chi ldren 's Guild, Ltd., DC Public Charter School Page 12 ( ( developmental disabilities and/or autism. Given The Children's Guild is the charter's educational operator, it will provide ongoing support to the CGDC faculty to train them in operating in accordance with the Individuals with Disabilities Act {IDEA}, designing responsive general education classes that provide access to the general education curriculum and accommodate students with special needs and support them to perform to high standards. Components of the school that address the anticipated needs of CGDC students will include implementation of the Transformation Education philosophy, TEACCH structures and philosophy, and methods of instruction such as Project Based Learning, workshop format, lesson design, a balance of whole group and sma ll group, fieldwork and outside experts, Arts Integration, and Information and Communication Technology {ICT}. The culture of CGDC is derived from TranZed philosophy that focuses on aligning the faculty's mindsets, the school's systems, its physical environment and its instructional approach to create consistent messages with the school's mission, beliefs and values. The culture resonates throughout the building and community sending messages about its purpose and function . Staff participates in daily Culture Card meetings to openly discuss the implementation and impact of the school's culture to align everyone's thoughts and actions to TranZed beliefs and values and apply them in their work with students and families. CGDC's Impact on the Surrounding Community ( The rationale for our school location was determined by two factors. First, we learned of the availability of a school building in Ward 7. The Maya Angelou Charter School is seeking a tenant to fill vacant space resulting from the pending closure of its middle school. We found the space and location ideal given it was an existing school, could accommodate 450 students and it was located in Ward 7. At the present time, CGDC does not have an alternative facility if the Maya Angelou School does not work out. However, CGDC is actively looking for additional facilities consistent with its vision to add a pre-K and a high school program . The closure of the middle school program at Maya Angelou School creates a need for middle school student seats which the CGDC could immediately serve. Therefore, a charter school of excellence would be wellreceived in Ward 7 and would assist it in its plans to stabilize and strengthen the avai lable educational options. It would also assist in supporting the financial stab ility of the Maya Angelou School through lease payments that will cover the cost of the vacant space resulting from the pending closure of Maya Angelou's middle school. Second, our meetings with Two Rivers School, Creative M inds School, Bridges, St. Coletta's and Naomi Rubin DeVeaux at the DC Charter School Board indicated The Children's Guild could have a positive influence on charter and DCPS schools in Ward 7 and all schools throughout the District by helping to meet the need for a school to serve a significant popu lation of children with special needs. In addition, CGDC will be a support to existing DC charter schools by helping them meet the needs of their special education population through training and consultation and providing a highly-skilled Diagnostic and Evaluation program. ( The Children's Guild, Ltd., DC Public Charter School Page 13 ( Third, Chris Tessone, the COO ofthe operating entity the See Forever Foundat ion, has indicated that he is will ing to assist in introducing The Children's Guild to, and gaining the support of, the neighborhood's leadership. Also, the mission of The Children's Guild, to serve socially and economically disadvantaged children and children w ith special needs, is compatible with the population served by the Maya Angelou School. c. Educational Focus Educational Focus The educational focus of CGDC is to meet the needs of every learner through a unique combination of researched best practices including Transformation Education, TEACCH and Project Based Learning, Ch aracter Development, Arts Integration and Information and . Communication Technology. CGDC will focus on serving specia l education student s in the District w ith in a comprehensive general education setting. A continuum of special educat ion services will be provided in addition to a Diagnostic and Evaluation program. CGDC' s educational focus is on creating an educational experience that fosters the ability to think critically, solve problems creatively, be self-disciplined and create caring students who serve a cause larger than themselves. To achieve its educational focus, CGDC will employ several key education practices including TEACCH, Project Based Learning, Brain-Compatible Instruction, Character Development and Arts Integration. TEACCH (Tra ining and Education of Autistic and Related Communication Handicapped Children) was developed at the University of North Carolina and provides the foundational structures that enable every student to succeed: environmental organization, schedules/routines, work systems and visual structures. Organizing the physical environment, developing schedules and work systems, making expectations clear and explicit, and using visual materials have been effective ways of developing skills and allowing students to work independently with success. Project Based Learning (PBL) is the instructional delivery system that identifies teaching strategies that connects the "what" = Common Core State Standards to the "how" = Project Based Learning. PBL will enable teachers to differentiate learning to address the diverse learning needs of the students whi le providing real world connections and application of the curricular concepts. Diagnostic and Evaluation (D & E) services of and for learn ing is a critical feature of CGDC. The D & E service specifically identifies the needs of students and precisely prescribes the strategies, interventions and mindset to maximize that student's success. This service would not be limited to CGDC stu dents. Other District charters may contract for this service as a means of improving services to se lected students without transferring those students out of their school. Within CGDC, diagnostic prescriptive services are ava ilable to assist the classroom team identify ( ( The Children's Guild, Ltd., DC Public Charter School Page 14 ( learning styles, strengths and weaknesses in order to best provide an instructional program that meets the needs of the student and is aligned with the District's learning outcomes. Character Development is also integrated throughout the content areas and aligns with the school's values. Students identified as requiring additional behavioral support will benefit from utilization of the Student Support Center (SSC). In the SSC, students develop an understanding of what gets in the way of their ability to comply with school-wide expectations and learn strategies to employ in the future. Teachers receive feedback and support from the SSC in how to effectively accommodate the learning/beh avioral needs of their students. Together these components contribute significantly to the culture of the school and ensure that every student is successful. Arts Integration embeds the arts within the core academic subjects to foster engagement, aide in retention and enhance comprehension . David Sousa, author of "How the Brain Learns" states: "Studies consistently show the following in schools where arts are integrated into the core curriculum: students have a greater emotional investment in their classes; students work more diligently and learn from each other; cooperative learning groups turn classrooms into learning communities; parents become more involved; teachers collaborate more; art and music teachers become the center of multi-class projects; learning in al l subjects becomes attainable through the arts; curriculum becomes more authentic, hands-on and project-based; assessment is more thoughtful and varied; and teachers' expectations for their students rise." Information and Communication Technology {ICT) provides opportunities for the enhancement of learning and may sign ificantly support students in their inquiries, and in developing their conceptual understanding. CGDC views technology as a tool for learn ing, albeit with its own set of ski lls, as opposed to an additional subj ect area. ICT skills shou ld be developed and learned in order to support the needs of individual learners in their inquiries. ( ( The Chi ldren's Guild, Ltd., DC Public Charter School Page 15 ( 2. Goals CGDC will elect not to use the PMF in setting goals. Instead, CGDC will set equally ambitious goals that measure student performance, school performance, and operational goals similar to those used at existing Children's Guild Monarch charter schools. Using these similar measures of success, CGDC will accurately measure and hold itself accountable for high student performance. CGDC will meet or exceed all goals set by the D.C. Public Charter School Board and will work with the Board to adjust our goals as deemed necessary. The goals of CGDC are: 1. To ensure that all students receive the support necessary to meet or exceed the curriculum standards and acquire the life skills necessary to be college and career ready. 2. To create a safe, academically and socially-rich environment that enab les students t o utilize creative expression, be self-d isciplined and make learning a life-long process. 3. To develop in students the mindset encompassing the values of caring, contribution and commitment to develop the skills of vision, courage and will and the understanding that the learning process entails struggle, transformation and enlightenment. 4. To partner with parents, guardians and the community as learning resources. CGDC Assessments: CGDC is committed to using PARCC and NCSC assessments to measure student achievement. As we gather data, we will revisit the measures indicated below to determine rigorous instruction . Indicator Metric Target ( Student Progress NWEA/MAPs Student Achievement PARCC Mean will meet or exceed the expected growth percentile at each grade -reading Mean will meet or exceed the expected growth percentile grade math Reading - 30% will achieve Proficient and Advanced Math - 30% will achieve Proficient and Advanced NCSC Gateway Fountas and Pinnell Attendance Leading Indicators 75% students will achieve Proficient and Advanced 75% of students will meet grade level benchmarks on running records Student attendance will meet or exceed 95%. ( The Children's Guild, Ltd ., DC Pub lic Charter School Pa ge 16 ( Systemic Evaluation Tools Available: • PBIS (a data and intervention tool) • Responsive Classroom Assessments • Student Support Team Assessment • START Team Assessment (refers to beginning of seeki ng solutions to problem behaviors) • Critical Skills Assessment • Positive Everyday Routines • School-Home Connection Assessment • ALSUP -Assessment of Lagging Skills and Unsolved Problems Continuous Quality Improvement Measures (CQI). Transformation Education deploys an evaluation system that ensures that objectives related to safe and orderly schools, enriched environments, culture deployment and fidelity, parent and community partnership development, effective leadership and academic growth are being met. CQI oversight is the responsibility of the Chief Operating Officer of Schools (COO) of The Children's Guild. The COO provides quarterly reports comprised of recommendations for continuous improvement to school administration and school leadership teams. Enrichment Class Survey Tools are utilized in physical education/life fitness, computer studies, music, visual and/or the dramatic arts. In each area the character traits of compassion, being respectfully open-minded, having ownership of one's behavior, being reflective, being international minded, being a risk-taker and displaying curiosity are taught and practiced. Environment Assessment Tools consist of teacher reflections, observation and feedback documents that focus on the teacher-learner relationship as a critically important element to the learning environment. The tools not only look at the psychological environment, that is evident in the characterization of the teacher-student relationship of trust, safety, and mutual respect, but also the physical environment and how it is fully and effectively used to teach CGDC's beliefs and values. Executive Function Assessments are tools that provide a general framework for assessing the extent to which executive skills are, or are not, being engaged consciously by the student . Therefore, executive functions skill application can be observed by teachers and therefore make it possible for the teacher to prompt the student to use the ski lls they have been taught to achieve a desired outcome. Fieldwork is a scheduled PBL activity that takes students out into the world to do st udies and investigations at various sites around the District. Fieldwork is a requirement for all students and is graded and counts towards a student's grade in every subject. Fieldwork provides students with the opportunity to collect authentic data and interact with experts to deepen their conceptual understanding. It also provides teachers opportunity to assess content and the school's character traits . The Children's Guild, Ltd., DC Public Charter School Page 17 ( ( ( Fountas' and Pinnell's is a reading enhancement program. Its goal is to support the student's development of self-initiating actions he/she will be able to apply to texts of similar difficulty. With daily teaching, the teacher helps the student climb the ladder of text difficulty with success. The focus of this guided reading approach is to bring the student up to the level of complex texts appropriate for the grade. To do so, teaching must begin with where the student is able to engage w ith some success, so that the student builds both confidence and motivation, thereby accelerating the student's development of a self-extending system for processing increasingly complex texts. Learning Targets form the foundation for all areas of assessment: assessment for learning; assessment of learning and; communication of assessment results . Quality learning targets play an integral role in using assessment to engage, support , and hold stude nts accountable for rigorous learning. Students regu larly self-assess against long-term and supporting targets and track their progress. Learning Walks are a series of focused classroom visits by school leaders that help leaders get a sense of the "big picture" of the school. The focus is on identifying patterns of practice in the building and providing leaders a structure for observation, documentation, conversation and reflection on instructional practice . ( Literacy and Math Workshops expose students to a variety of instructional strategies and work on a specific set of predetermined learning targets through mini-lessons, guided or independent work time, and one-to-one/small group conferences. At the end of each workshop, assessment opportunities are available to teachers through the sharing of student work, and through brief critique sessions that provide students with substantive feedback from their peers and teachers. Measures of Academic Progress (MAP) are computerized, adaptive assessments in reading and math (with the option of science) and will be administered in October, February, and May of each calendar year. Students in kindergarten and grade one take the primary form of the test. All tests are aligned to the Common Core Curriculum and identify the instructional level of each student and measure growth over t im e. The Parent Satisfaction Survey/Student Satisfaction Survey is given out annually and evaluates the school's performance from a parent and a student perspective. The Parent Advisory Group works with the schoo l's leadership team to develop a strategic plan with the goal of 100% parent/student satisfaction . For copies of th e Parent Satisfaction and Student Satisfaction Surveys CGDC will use, see Appendix, Section K. Passage Portfolios are created by students at the end of grades five and eight to demonstrate the student's mastery of grade-level learning targets. Each passage portfolio is standards-based and provides work samples as evidence of learning target mastery in reading, writing, math, ( The Children's Guild, Ltd ., DC Public Charter School Pa ge 18 ( science, and social studies. Work included in the portfolio must meet or exceed grade level expectations for the required learning targets . A rubric is used to assess portfolios. Presentations of Learning (POL) is a promotion requirement for students in grades S-8. As part of the POL, students participate in a culminating project. The project requires that each student demonstrates engagement with the five essential elements of the program : knowledge, concepts, skills, attitudes and action. The purpose ofthe POL is to share the student's work with a panel made up of parents, administrators, invited guests, relatives and friends. Participation is mandatory and preparation of a POL is a promotion requirement. POL is a trans-disciplinary inquiry conducted in the spirit of personal and shared responsibility, as well as a summative assessment activity that is a celebration as students move from the elementary years into the middle years and from the middle years into high school. A rubric is used to assess the POL. Project-Based Assessments. Projects demonstrate mastery of skills or completion of specific tasks. A project is a compendium of complex assignments, each directed toward a common goal. Projects are designed to address learning targets and are scored by using a rubric, which is shared with students in advance. Project-Based Assessments may include performance assessments which are judged according to pre-established performance criteria as well as the use of exemplars, check-lists, anecdotal records and continuums. A variety of assessment strategies such as process-focused assessments, selected responses and open-ended tasks are also used. ( Scholastic READ 180 and System 44 Next Generation are reading interventions used for students performing significantly below grade equivalent peers. READ 180 is a data-driven reading intervention program, which means student performance immediately impacts instructional approach . A variety of formative assessment instruments identify students' most urgent needs, enabling the program and the teachers to adjust instruction accordingly. READ 180 and System 44 are based upon a blended learning model that incorporates whole group instruction, small group in struction and computer based instruction. Teachers can differentiate instruction ba sed on the learners' needs in addition to the dashboard progress monitoring provided through the program . These programs are recognized as effective systems for providing rigorous reading and writing expectations that align with cess. Service Learning Log. Service learning is an integral part of many learning units and projectbased experiences. It provides an authentic community need. Each year, students participate in at least one unit of study that includes service learning. This experience is captured within their Service Learning Log. Student Learning Profiles are con structed by teachers from the MAP data which provides a basel ine for assessment. They are used to diagnose inst ruct ional needs and make data-driven decision s to support academic growth. Students who are high achieving will also be identified so that teams can structure higher level thinking activities and extended learning activities suitable to th e academic capabilities of each child. ( The Children's Guild, Ltd., DC Public Charter School Page 19 ( Student-Led Conferences are 20-minute student presentations to parents that include an introduction, a summary of a successful learning target in reading and math, and an exp lanation of a strong piece of written work. In addition, students explain the project-based unit of study that led to their culminating project from social studies and science. Parents and teachers give students feedback on the presentation and evaluate quality of student reflection and representation of their thinking. Student Portfolio Assessments. Student portfolios are a collection and communication device. They are used to communicate visually about the student's talent, style, and range of work. The contents are selected to offer a rich and detailed view of the student's academic and social development characteristics and qualities. Because students are involved in the coll ecting, interpreting and sharing of portfoli o content, students take notice of, keep track of, and celebrate their learning They become reflective learners through the portfolio internal feedback loop, learn to set goals, and recognize that competencies and cha llenges are all habits of thought cultivated by the portfolio assessment process. The Culture Card is an effective tool for ensuring there is a clear and consistent message describing our organizational culture (beliefs, values, mission, workplace norms and wisdom principles). See Appendix, Section L. CGDC w ill use this system to teach how the organization app lies the wisdom principles and its foundational beliefs in the workplace. ( Community Circles meet on a daily basis in every grade. This structure allows for relationship building, review/discussion of school-wide behavioral expectations, daily schedule and Character Development. Classroom issues can be discussed and resolved as a means of creating an atmosphere of collaboration and an absence of fear. Behavior Motivation Systems and Procedure Maps are part of a comprehensive program developed by The Children's Guild to encourage students in our charter schools to select behaviors consistent with learning and to become responsible citizens in the learning community. These systems and tools support an integrated and aligned system of beliefs, mindsets, structures and processes designed to create in students the desire to act con sistent with CGDC's values and to develop in students the skills to execute those actions. CGDC will use the resu lts of cohort analysis to bring focus to whether goals and performance standards are being met. By the end of 2015-2016, the CGDC leadership team will determine patterns of student achievement using the results of PARCC testing and MAP growth reports (and annually thereafter) . By the end of 2016-2017, the CGDC school leadership team will also use the resu lts of parent, student and faculty satisfaction surveys and Project Based Learning implementation review data to determine whether the desired outcomes of the program goals have been met (and annually thereafter) . In addition, the leadership team will disaggregate the data with careful attention to student sub-groups to determine achievement gap patterns. When patterns emerge, it is the responsibi lity of the leadership team to address whether pedagogical or curricular changes need to occu r. It is also the leadership team's responsibility to determine whether tiered intervention s are meeting the individual needs of students. Data The Children' s Guild, Ltd ., DC Public Charter Schoo l Page 20 ( ( will be collected and stored through various form ats and software/online programs. CGDC is exploring PowerSchool software for attendance monitoring/reporting, scheduling, assignments, grades and communication with teachers. NWEA, PARCC, SWIS and READ 180/System 44 are on line assessments. Spreadsheets will be developed for student support data analysis. Grade level data teams w ill analyze data weekly and the Student Support Team will meet bi-monthly to review and ana lyze student performance and behavioral data. The leadership team will use these multiple data sources on a quarterly basis as part of a continuous quality improvement plan (CQI) in determining how CGDC is progressing in student performance, attendance, behavior and overall program outcomes. The leadership team will prepare an outcome data sheet template as part of its reporting practices as well as any additional record keeping required by DCPS. CQI Summaries and data will be shared with the CGDC Board of Directors and the DCPCSB. ( ( The Children's Guild, Ltd., DC Public Charter School Page 21 ( CQI Process Continuous Quality Improvement (CQI) is a philosophy and a quality management process that encourages charter school team members to continuously self-assess by asking the questions, "How are we doing?" and "Can we do it better?" To address these questions, a practice needs structured data from a variety of school dimensions to review and analyze (academic, cultural, program implementation. engagement , etc ...) Quest: To continuously improve the quality of education at CGDC through focused and persistent implementation of the school's mission and vision. ccomplish? (Work plan development, set a cad conditio , g als & set Learning Targets for Staff) How will we know t at a chan e is an improvement? (Data collection What changes can w result in improvements? ( As described above, the CGDC leadership team will use multiple data sources on a quarterly basis to track and assess progress toward goals for students with disabilities. In addition to the data sources identified for the genera l education students above, special education classrooms will also utilize the following data sources: o Individualized Education Plans. Each quarter the student's progress will be assessed and identified on the IEP. Progress monitoring of goals and objectives will be reviewed indicating if the student is making progress, not making progress or regressing. A copy of the IEP goals and objectives will be sent home quarterly. The Children's Guild, Ltd., DC Public Charter School Page 22 ( o o Functional Behavioral Assessment/Behavior Intervention Plan (FBA/BIP), Each quarter the student's progress will be monitored in alignment with the FBA/BIP. School Wide Information System (SWIS). Student behavioral progress will be monitored monthly through SWIS to identify trends in behavioral incidents and determine if additional interventions are warranted. o READ 180/System 44. Student performance is monitored using the Scholastic Reading Inventory online assessment in addition to the teacher dashboard identifying daily student performance. o o Math. CGDC is currently exploring remedial math resources such as Fast Math, Understanding Math and EnVision Math Diagnosis Intervention Systems. Team Primacy Meetings. These meetings occur weekly to review and analyze data, identify modifications needed, access effectiveness of interventions (academic and behavioral), identify resources needed and develop plans of action to address deficit areas. CGDC will evaluate special education students' performance towards IEP goals and objectives quarterly. Documentation of the student's progress will be completed through the online DC IEP process. Student progress will be reviewed quarterly, annually, and as requested, through the IEP annual review process in collaboration with the DC special education case manager. ( ( The Children's Guild, Ltd ., DC Publ ic Charter School Page 23 ( 3. Charter School Curriculum a. Student Learning Standards The criteria identified for the selection of standards and curricular materials is derived from the goals of CGDC. Given the student population CGDC proposes to educate, the standards and materials se lected must be highly adaptable to meet the learning needs of very diverse learners, be literacy infused, contain a technology component to allow for varied presentation of learning, inquiry-based to support project-based learning strategies and common-coreinformed to align with the standards and the PARCC assessment. A history of 60 years of effectively meeting the needs of special education students in nonpublic specia l education settings provides CGDC with a wealth of resources to effectively meet the needs of all learners. The structures of TEACCH combined with the components of Project Based Learning and Transformation Education provide a powerful and effective framework for supporting the learning needs of each student. English Language Arts CGDC will implement the Common Core State Standards (CCSS} in English Language Arts. The cess are designed to be robust and relevant to the real world, reflecting the knowledge and ski lls that CGDC students need to succeed in college and careers. The standards are designed to ensure that students graduating from high schoo l are prepared to enter college or the workforce. The cess promote equality, by ensuring all students are prepared with skills and knowledge necessary to collaborate and compete with their peers in the United States and abroad. The CCSS will ensure more consistent exposure to materials and learning experiences through curriculum, instruction, and teacher preparation among other supports for student learning. In a global economy, students must be prepared to compete with not only their American peers in the next state, but with students from around the world. These standards will help prepare students with the knowledge and skills they need to succeed in college and careers . Curriculum materials will be literacy-infused with an emphasis on informational text to align with the cess. Materials will be highly adaptable to accommodate the various learner styles and abi lity levels within the student body. ( ( CGDC will use the Common Core State Standards for math. These standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in students. These practices rest on important "processes and proficiencies" with longstanding The Children's Guild, Ltd., DC Public Charter School Page 24 ( importance in mathematics education. The process standards come from National Council of Teachers of Mathematics and include problem so lving, reasoning and proof, communication, representation and connections. The National Research Council's report provides the mathematical proficiency strands of adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures, flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensib le, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy). Science CGDC will implement Next Generation Science Standards (NGSS). The scientific practices in the NGSS, as defined by the National Research Council (NRC), include the critical thinking and communication skills that students need for post-secondary success and citizenship in a world fueled by innovations in science and technology. NGSS is based on research exploring how students learn science which is complementary to TranZed's brain-based learning approach. The NGSS framework describes a vision of what it means to be proficient in science; it rests on a view of science as both a body of knowledge and an evidence-based model and theory building enterprise that continually extends, refines, and revises knowledge. It presents three dimensions that are combined to form each standard : (1) practices; (2) crosscutting concepts and; (3) disciplinary core ideas. NGSS reflects the interconnected nature of science as it is practiced and experienced in the real work. This aligns with components of Project Based Learning whereby students .explore essential questions found in the real work through integrated curricular concepts to effectively problem solve the question. Social Studies CGDC will use the Common Core Literacy Standards in addition to the District of Columbia Social Studies Standards. While the standards present topics in a chronologically organized pattern, teachers will be encouraged to elaborate on topics and enrich the learning experience by incorporating current events. This aligns with our Project Based Learning approach which entails using research (historical perspective) to solve current world issues. The Social Study Standards are based on a solid base of factual knowledge which supports critical thinking. The central ideas, events, people and works that have shaped our world, are critical for our students to remember and understand . In addition, the standards necessitate that students acquire a firm grasp of reasoning and practice in inquiry and research. Students must learn how to frame and test hypotheses, distinguish logical from faulty reasoning, frame reasoned options and arguments, and grasp reflective thinking and evaluation . The Arts. CGDC will use the K-12 National Arts Education Standards (NAES) that outline what every K-12 student should know and be able to do in the arts. The NAES were developed by the Consortium of National Arts Education Associations, through a grant administered by The The Children's Guild, Ltd ., DC Public Charter School Page 25 ( ( ( National Association for Music Education (NAME). Th e NAES are developmentally appropriate as they expect that students will develop high er level s of the required skills and knowledge through being exposed to increasing complex works of art and that stud ent responses to the art will be increasingly more sophisti cated. The NAES framework promotes th e students' critical thin king, working, communicating, reasoning, and investigating skills and provides for students' growing familiarity with the ideas, concepts, issues, dilemmas, and knowledge important in the visual arts. As students gain knowledge and skill, they expand their ability to apply the knowledge and skills learned through the arts in their widening personal worlds. Additional Academic Area(s) English Language Learners The National Governors Association Center for Best Practices and t he Council of Chief St ate School Officers strongly bel ieve t hat all students should be held to the same high expect at ion s outl ined in the Common Core State Standard s. This includes students who are English language learners (ELLs). However, these students may require additional time, appropriate in st ructional support, and aligned assessments as they acquire both Engli sh language proficiency and content area knowledge. Stand ards are attainable when English language learners have the ability to access content. CGDC's project-based instructional approach extends the time students have for gettin g language support services while giving th em a jump-start or pre-loading on content subjects. CGDC t eachers will offer multiple pathways for students to demonstrate their understanding of the content . Assurance of ELL access is addressed at many levels of in struction as the followi ng table illust rates : Preparation Clearly defined content objectives Clearly defined language objectives Content concepts appropriate for age and educational background level of student Supplementary materials used to a high degree, making the lesson clear and meaningful (e.g., graphs, models, visuals) Adaptation of content to all levels of student proficiency Meaningful act ivities that integrate lesson concepts (e.g., surveys, letter writing, simulat ions)wit h language pract ice opportunit ies for reading, writing, listening and/or speaking) Building Background Concept s explicitly linked to students' background experiences Links explicitly made between past learning and new concepts Key vocabulary emphasized Comprehensible Input Speech appropriate for students' proficiency level(e.g. slower rate and enunciation, and simple sentence structure for beginners) Explanation of academic tasks are clear Uses a variety of techniques to make content concepts clear Strategies Provides ample opportunities for students to use strategies Consist ent use of scaffolding techniques throughout lesson, assisting and supporting student understanding (e.g. t hink-alouds) Teacher uses a variety of question types, including those t hat promote higher-order thinking skills ( ( Interaction Frequent opportunities for interaction and discussion between t eacher/student and among studens, which The Children's Guild, Ltd., DC Public Charter School Page 26 ( Practice/Application encourage elaborated responses about lesson concepts Grouping configurations support language and content objectives of t he lesson Consistently provides sufficient waiting time for student responses Ample opportunities for students to clarify key concepts Provides hand-on materials and/or manipulatives for students to practice using new content knowledge Provides activities for students to apply content and language knowledge in the classroom Uses activities that integrate all language skills Lesson Delivery Content objectives clearly supported by lesson delivery Language objectives clearly supported by lesson delivery Students engaged approximately 90% to 100% of the period Pacing of the lesson appropriate to the students' ability level Review/Assessment Comprehensive review of key vocabulary Comprehensive review of key content concepts Regularly provides feedback to students on their output (e.g., language, content, work) Conducts assessment of student comprehension and learning of all lesson objectives (e.g. spot checking, group response) through-outthe lesson ( ELL students will bring with them many resources that enhance their education and can serve as resources for CGDC and the community. Many ELLs have first language and literacy knowledge and skills that boost their acquisition of language and literacy in a second language. Additionally, they bring an array of talents and cu ltural practices and perspectives that wi ll enrich our school and wider community. CGDC teachers will build on this enormous reservoir of talent and provide those students who need it with additional time and appropriate instructional support. Thi s includes language proficiency standa rds that teachers can use in conjunction with the ELA standards to assist ELL students in becoming proficient and literate in English. To help ELL students meet high academic standards in English Language Arts, it is essential that they have access to: • • • • Teachers and personnel who are well prepared and qualified to support ELL students while ta king advantage of the many strengths and skills they bring to the classroom; A literacy-rich school environment where students are immersed in a variety of language experiences; Instruction that develops foundational skills in English and enables ELL students to participate fully in grade-level coursework; Coursework that prepares ELL students for post-secondary education or the workplace, yet is made comprehensib le for students learning content in a second language (t hrough specific pedagogical techniques and additional resou rces); Opportunities for classroom discourse and interaction that are designed to enable ELL students to develop communicative strengths in language arts; Ongoing assessment and feedback to guide learning; and Spea kers of English who know the language well enough to provide ELL students with models and support. • • • Students with Disabilities ( CGDC will establish a culture of in clusiveness that will ensure that all students have access to the general ed ucation curriculum . Students with disabilities will be included within the general The Children's Guild, Ltd ., DC Public Charter School Page 27 ( education curriculum based on the cess as well as other school activities to the greatest extent possible. It is both a legal requirement and moral obligation to educate students with disabilities alongside their general education classmates to the fullest extent possible. The Children's Guild has an extensive history of successfully educating students with specia l needs in addition to charter schools that have successfully implemented a continuum of specia l education services. Students with disabilities - students eligible under the Individuals with Disabilities Education Act {IDEA}-must be challenged to excel within the general curriculum and be prepared for success in their post-school lives, including college and/or careers . These common expectations provide an historic opportunity to improve access to rigorous academic content standard s for stude nts with disabilities. The cont inued development of understanding about research-based instructional practices and a focus on their effective implementation will help improve access to mathematics and English language arts {ELA} sta ndards for all stu dents, including those w ith disabilities. In order for students with disabilities to meet high academic standards and to fully demonstrate their conceptual and procedural knowledge and skills in mathematics, reading, writing, speaking and listening {English Language Arts}, their instruction must incorporate supports and accommodations, including: • Supports and related se rvices designed to meet the unique needs of these students and to enable their access to the general education curriculum {IDEA 34 CFR §300.34, 2004}. • An Individualized Education Program {IEP} which includes annual goals aligned with and chosen to facilitate their attainment of grade-level academic standards. • Teachers and specialized in structional support personnel who are prepared and qualified to deliver high -quality, evidence-based, individualized instruction and support services. Promoting a culture of high expectations for all students is a fundamenta l goal of the Common Core State Standards. In order to participate with success in the general curriculum, students with disabilities, as appropriate, may be provided additional supports and services, such as: • Instructional Supports for Learning- based on the principles of Universal Des ign for Learning {UDL} - which foster student engagement by presenting information in multiple ways and allowing for diverse avenues of action and; • Instructional Accommodations {Thompson, Morse, Sharpe & Hall, 2005} - changes in materials or procedures which do not change the sta ndards but allow students to learn with in the framework of the cess. • Assistive technology devices and services to ensure access to the general education curriculum and the cess. Some students with the most significant cognitive disabilities will require substantial supports and accommodations to have meaningful access to certa in standards in both instruction and assessment, based on their communication and academic needs. These supports and The Children's Guild, Ltd ., DC Public Charter School ( Page 28 ( accommodations should ensure that students receive access to multiple means of learning and opportun ities to demonstrate knowledge, but retain the rigor and high expectations of the Common Core State Standards. CGDC will provide a continuum of tiered interventions in accordance with Response to Interventions (RTf) initiative. Tiered instruction delivers instruction to students on various levels related to the nature and severity of the student's difficulties. This differentiation allows students at risk to meet academic expectations. Tier 1 is regular classroom instruction, different iated as much as possible within the class room to meet the individual students' needs. We anticipate approximately 80% of the students will succeed in this t ier as they are achieving the learning targets prescribed in the cess at the proficient or advanced level. The core reading and math curriculums high quality instructional programs implemented in alignment with the cess. Tier 2 instruction is characterized by small group instruction (composed of three to six students), meeting three to four times each week for 30 to 60 minutes each, for nine to twelve weeks . We anticipate that approximately 15% of the students will be served by Tier 2 interventions. The needs of t hese students are identified through the assessment process and instructional interventions are delivered from the focus of the specific targeted needs. Remed iation of the targeted skill is provided in the small group setting using supplemental interventions identified for students at risk of not meeting academic standards. Tier 3 instruction is characterized by one-to-one or sma ll group instruction (for one to three students), meeting daily for 30 to 60 minutes each, for nine to 12 weeks. We anticipate that Tier 3 students will be approximately 5% of the student population. Tier 3 students are considered to be at high risk for failure and if not responsive to an intensive, specific intervention may be candidates for identification as having special education needs. Supplemental interventions for students at high risk are provided at this level. In addition to the continuum of instructional interventions provided through the tiered system, TEACCH methods w ill be used throughout the school and in particular with students w ith disabilities. TEACCH is an evidenced-based program that supports all learners. The developers of TEACCH based their model on the idea that to effectively teach students (particularly those with Autism) a teacher must provide structure. Although this method was originally intended for use with students with autism spectrum disorders, it can be adapted and used in any educational setting. The Children's Guild effectively implements the components of TEACCH with their specia l education schools across all grade levels and disabilities. Four essential components of the TEACCH philosophy will be utilized to help students access the curriculum in meaningful ways . The arrangement of the physical environment is important for increasing understanding by reducing anxiety which directly impact's the student's behavior and ability to learn. A clear, consistent and predictable physical environment enables students to develop the organizational skills needed to access the curricu lum across subjects. The work The Children's Guild, Ltd., DC Publ ic Charter School Page 29 ( ( ( areas should be clearly marked so students can independently find their way to the different locations within the classroom to perform the learnin g activity associated with that location. This promotes independent learners who can access learning without adult support. The second essential component is the use of visual schedules. These visual schedu les help inform students of what occurs in their day and in what sequence events will happen. By reducing confusion and increasing flexibility, students have great success with accessing activities throughout their day. The daily schedu le also provides information for staff about which teacher(s) and/or aides will be in which areas. Schedules also help with transitions by preparing students for what comes next and eventually leading to independence. The daily schedule shou ld balance opportunities for individual, independent, group and leisure activities throughout the day. A work system is the third component of the TEACCH model. Individualized work systems help keep students organ ized so they can function independently and effectively in a variety of different activities. Work systems include where materials are located, the order in which activities get completed and where students place work when it is finished. Students learn how to make choices and develop personal autonomy through individualized work systems. Visual structures are the fourth component of the TEACCH model. Visual structures are used to differentiate tasks for students by highlighting, organizing and clarifying important and relevant information. Visual structures can also identify where the student's materials and/or desk shou ld be placed, where a student can move throughout the room or how information should be presented on the paper. Carefully assessing the visual structures needed by each student to access the curriculum will enable him/her to be successful. Diagnostic and Evaluation Services As a highly-skilled, unique educational and behavioral assistance tool, CGDC will provide a diagnostic and evaluation (D&E) program within the charter school. The rationale for placement of this D&E within the charter school is to better compare the behavior and thinking of the identified student with students not identified as requiring a restricted setting. This allows staff the opportunity to observe the student in the context of a school setting (classroom, hallway, cafeteria, playground, socia l situations, etc) and determine the lagging ski lls that impact the student's ability to be successfu l in a less restrictive setting. This program will assist CGDC, and other charter schools within the District who refer students for placement in the program, in identifying the educational and social emotional needs of a student in order to effectively place the student in the right educational program. The D&E process will be integrated into CGDC through the school's Student Support Team and leadership team to address the specific needs of each student identified for the program. Upon completion of the D&E process, a diagnostic summary is w ritten identifying the strengths and weaknesses of the student and contains a learning profile outlining specific instructional modifications, intervention strategies and placement recommendations to effectively meet the students' educational and behavioral needs. ( ( The Children's Guild, Ltd., DC Public Charter School Page 30 ( CGDC expects the student population to need targeted interventions; rigorous learning expectation s with structures and processes supporting student progress; and high engagement with opportunities for student leadership which builds ownership of learning and drives achievement. The CGDC's methods of instruction are derived from the mi ssion, be liefs and values of the school and prepare the student with college and career readiness skills. These goals include student engagement in the learning process, development of new skills, prevent ion of behavior problems through proactive intervention strategies, development of critical th inking skill s, the ability to problem solve creatively, development of self discipline and a commitment to serve a cause larger than oneself. Evidence of achieving these goals is reflected by the student's ability to demon strate both academic and behavioral progress. CGDC's teaching strategies are the vehicle by which the instructional goals and schoo l outcomes become actua lized. The Common Core State Standards are the curriculum or "what" that in struction is based upon and Project Based Learn ing (PBL) is "how" students become engaged with the curriculum . PBL is based upon the inquiry process and incorporates Bloom's taxonomy higher level ski lls: create, evaluate, analyze and apply. Centered on a real world problem, PBL incorporates creative problem so lving, researching potential so lutions, interfacing with various forms of technology and the creation of a product that demonstrates student learning. The PBL products/projects provide opportunities for art integration as children integrate cross curricu lar concepts into a finished product that may include a performance, music, art or techno logy. The TEACCH principles of visual structures, schedules, work systems and physical organization provide the foundation for answe ring the 5 basic questions students have about the learning experience: 1. 2. 3. 4. 5. Where am I going? What am I going to do when I get there? How much do I have to do? When will I be finished? What will I do next? ( Neuroscience has shown that when there is an absence of fear and students understand what is expected of them, they increase their cognitive availability for learning. As a result , students can function independently within the school environment when they have a clear understanding of what is expected of them . TEACCH al so emphasized the importance of incorporating the students' strengths when teaching new skills. Teachers integrate the students' strengths when they make real work connections and help students apply what they are learning in the classroom with their community. Technology is infused throughout the instruction al process allowing students to understand the global economy in addition to providing an accommodation for students with disabilities. The Children's Guild, Ltd., DC Pub lic Charter School Page 31 ( b. Resources and Instructional Materials English Language Arts: Curriculum materials will be literacy infused with an emphasis on informational text to align with the cess. Materials will be highly adaptable to accommodate the various learning styles and mastery levels within the student body. Fountas and Pin nelli s is founded on a comprehensive approach that involves high impact interventions for struggling readers. Th is allows for differentiated instruction through working with small groups in reading. Leveled readers enable the teacher to provide relevant text for both the struggling reader and the accelerated reader. Guided reading provides the student with opportunities for explicit teaching to build his/her network of effective problem solvi ng skills. Benchmark assessments, in addition to observation, provide progress monitoring to determine the appropriate level for each student. Teachers use the text for children to expand what they know to do as readers . Daily 5 is a reading intervention that allows for differentiation, is an integrated literacy instruction and classroom management system and teaches students independence. It is based on brain research to identify the best t ime to introduce new concepts, to maximize student attention and directly link instruction to the needs of the students. Daily 5 has various instructional delivery formats so it is adaptable to the needs of the classroom . ( Scholastic READ 180 and System 44 Next Generation are reading interventions used for students performing significantly below grade equivalent peers. READ 180 is a data-driven reading intervention program, which means student performance impacts instruction. A variety of formative assessment instruments identify students' most urgent needs, enabling the program and teachers to adjust instruction accordingly. READ 180 and System 44 is based upon a blended learning model that incorporates whole group instruction, small group instruction and computer based instruction. Teachers can differentiate instruction based on the learners needs in addition to the dashboard progress monitoring provided through the program . These programs are recognized as an effective system for providing rigorous reading and writing expectations that align with cess. Accomplishing School Mission and Goals: These literacy resources support the teaching of a complex process of making meaning. In CGDC, comprehension strategies and critical thinking skills are taught K-8 to support students making sense of content and the world around them . Students " learn to read" while "reading to learn" from text in an increasing complex mannerpresenting an opportunity for students to go beyond their perceived limits and accomplish more than they thought possible while also preparing them for college and career success. Through our chosen instructional design and teacher differentiation, access to learning is provided to all students. ( The Children's Guild, Ltd., DC Pub lic Charter School Page 32 ( Math: Investigation Math curriculum is aligned to the CCSS for Mathematics and aligns with the goals and educational focus of CGDC. The Investigation Math curriculum is adaptable to meet the needs of all learners through interactive whiteboard activities, use of manipulatives to provide visual, concrete examples, differentiation and intervention guides, Spani sh components and software/technology. Implementation of the program includes whole group, small group and individualized instruction . The materials in Investigation Math help teachers support the range of learners in his/her classroom, provides regular opportunities for students to discuss important mathematical ideas and to review and practice them and encourages parents to learn about the curriculum and their child's th inking. Thi s aligns with CGDC's view of parents as partners and CGDC will provide Parent University workshops to assist them in understanding how their child's brain learns . Investigation Math uses routines for learning which is emphasized through the TEACCH methodology and brain-based learning. Research shows that when routines are implemented the brain is free to focus on new information. Creating a learning environment with predictable routines maximizes the student's ability to fully engage in the learning process. Accomplishing School Mission and Goals: CGDC math resources emphasize critical thinking skil ls in the areas of arithmetic, word problems, and other mathematical foundations. They also allow for a focus on big mathematical ideas, high quality student work, and allow for connections to the Units of Study and to teaching math in isolation. CGDC math teachers invite students to find patterns and relationships, to become flexible problem-solvers, to articulate their reasoning, and to reflect on their choice of strategies. Teachers cultivate mathematical habits of mind and often conduct class as a workshop . A workshop begins with a complex problem and continues with independent or group work, a mini-lesson based on what students are struggling with or have discovered, sharing/comparing problem-solving strategies and a synthesis of the day's learning. This sequence en sures that students are doing the thinking. Through our chosen inst ructional design and teacher differentiation, access to learning is provided to all students. ( ( Science: Our science curriculum will be researched best practices, as defined by the National Research Council and include the critical thinking and communication skills that students need for postsecondary success and citizenship in a world fueled by innovations in science and technology. Next Generation Science Standards (NGSS} are based on research to explore how students learn science wh ich complements TranZed's brain-based learning approach . The NGSS framework describes a vision of what it means to be proficient in science; it rests on a view of science as both a body of knowledge and an evidence-based, model and theory building enterprise that continually extend s, refines, and revises knowledge . It presents three dimensions that will be combined to form each standard: (1} practi ces; (2} crosscutting concepts and; (3} disciplinary core ideas. NGSS reflects the interconnected nature of science as it is practiced and experienced in the real work. This aligns with components of Project Based Learning whereby students explore essential questions found in the real work through integrated curricular concepts to effectively problem solve the question . The Children' s Guild, Ltd., DC Public Charter School Page 33 ( Reading, writing, thinking and working as scientists is a central focu s of instruction at CGDC. CGDC wants to develop students who see themselves as stewards of the natural world . Students engage in an interdisciplinary approach; connecting science, math, engineering, and technology while also building skills in quest ioning; developing and using models; planning and carrying out investigations; collecting; analyzing; and interpreting data; constructing exp lanations; designing solutions; engaging in argument from evidence and synthesizing and communicating information. Appropriate science resources will be researched in the planning year of the school development. The EQuiP NGSS Rubric is a tool for educators and education lead ers to use in identifying high quality, NGSS-aligned instructional materials through a criterion-based, peerreview process. This tool will be available in the first quarter of 2014, which will ass ist us in identifying materials that align with CCSS. Delta Education' s manipulatives will be utilized for science experiments in addition to use of websites fo r virtual labs and research on science concepts. Informational text will also be implemented in science to accordance with the science literacy standards and literacy in history/social studies and science and technology standards . Accomplishing School Mission and Goals: The variety of CGDC science resources provides the ( necessary f lexibility to building a culture of science inquiry and to utilize tran s-disciplinary skills. Through our chosen in structional design and teacher differentiation, access to learning is provided to all students. Social Studies: Our Social Studies curriculum will align with our Project Based Learning approach which entails using research (historical perspective) to solve current world issues. The curriculum will be inquiry based and adaptive to support the needs of diverse learners. The social studies curriculum is based on a solid base of factual knowledge which supports critical thinking. The central ideas, events, people and works that have shaped our world and are critical for our students to remember and understand are emphasized . In addition, the curriculum requires that students acquire a firm grasp of reason ing and practice in inquiry and research . Students must learn how to frame and test hypotheses, dist inguish logical from faulty reasoning, frame reasoned options and arguments, and grasp reflective thinking and evaluation {DC Social Studies Pre-K through Grade 12 Standards). Social Studies at CGDC support students in appreciating and understanding diverse cultures and understanding connection s among ancient and modern cultures and enduring themes through time. Students analyze primary resources, consider multiple perspectives, conduct research and draw thei r own conclusions. Literacy instruction is a focus at all grade levels in this content area as students learn to read, write, and think as historians. CGDC is intentional in creating a culture of social studies inquiry through its choice of resources and pedagogy. Through our chosen instruction al design and teacher differentiation, access to learning is provided to all students. Accomplishing School Mission and Goals: ( The Children's Guild, Ltd., DC Pub lic Charter School Page 34 ( Curriculum Development Timeline In the first year of CGDC, faculty will use existing curriculum maps and project based unit plans designed by The Children's Guild's other charter schoo ls that align with the CCSS. Throughout the school year, faculty will review and revise the curriculum maps in accordance with student achievement data. Lesson plans will be turned in weekly to the school's leadership team. Curricular maps include project based units, methods of instruction, lesson plans, resources, proposed fieldwork, experts and product development. Future curriculum maps will be developed by CGDC teachers through professional development using analysis of student achievement data. During the summer of 2014, the leadership teams of Monarch Academy Charter Schools and CGDC will collaborate to revise and develop curriculum maps in alignment with cess. This will allow sufficient time to review and purchase curriculum resources and materials that support CGDC's newly developed and/or revised curricular maps and cess. In addition, as student enrollment occurs, staff will be able to identify curricular resources to meet the needs of the student population. c. Methods of Instruction ( Teachers at CGDC w ill organize instruction using the following methods: 1. Project Based Learning- Integrated Units where content, knowledge and skills from separate subject areas are intentionally connected through cross-curricular investigations. Project Based Learning is the instructional delivery system that identifies t eaching strategies to connects the "what" = Common Core State Standards to the "how" = Project Based Learning. PBL will enable teachers to differentiate learning to address the diverse learning needs of the students while providing real world connections and applications of the curricular concepts. The essential elements of Project Based Learning include: o Focus on significant content o o o o o o o Develop 21st Century competencies Engage students in in-depth inquiry Organize tasks around a driving question Establish a need to know Encourage voice and choice Incorporate revision and reflection Include a public audience ( Another essential feature of the Transformation Education design for school improvement is a school-wide focus on Character Development and teamwork. The founders of CGDC have The Children's Guild, Ltd., DC Public Charter School Page 35 ( established the core values of carin g, contribution and commitment which will be integrated throughout our school culture, but specifically within the integrated units of study and projects. CGDC wants students and staff to show an understanding of others by treating others with kindness, compassion, generosity and a forgiving spirit. Caring does not exist without action. We want students and staff to contribute positively to the school community. As students and staff become part of the CGDC community, it is this commitment that will assist us in reaching both our program and educational goals. 2. Workshop format to introduce and explicit ly teach concepts, skill s, and strategies related to the learning targets. ( Workshop Model Instruction describes a set of structures, groupings, and activities that incorporate active pedagogy pract ices and protocols. Based on the premise that students learn by doing, this model ensu res that students get tim e each day to engage meaningfully w ith the subjects they are exploring. The workshop model provides a format for exposing students to a variety of instructional strategies as they work individually or in small groups on a specific set of predetermined learning targets, but it really allows students to take ownership of the learn ing process. Students at CGDC wi ll be in a literacy workshop of 70 to 75 minutes (depending on grade) and a math workshop of 70 to 75 minutes each day. In addition t o t hose workshops, instruction during the acceleration and enrichment periods will often use t he workshop model. Although no two workshops look exact ly alike, th e following components are typically present in some form : • • • • ( Mini-lessons. Organized around a clearly-focused objective that addresses a single elem ent of the work that stud ents w ill be doing, mini-lessons can be as short as five minutes or as long as 15 minutes. Mini-lesson s provid e explicit instruction in the skills that students will practice and apply during the workshop. They typically include mod eling and th ink-a louds. Guided or independent work time . Students generally use the guided and independent work time to practice, extend, and ref ine the skills addressed in the dai ly mini-lesson . Guided or independ ent work time takes up the bulk of time in any workshop. Well arti culated and consistently enforced classroom procedures, tradition s, and rituals, and established accountability systems help ensure that stude nts use this time well . One-on -one/small group conferences . Along with mini-lessons, conferencing provides a venu e for direct instruction during th e workshop. While the rest of the stud ents are working independently, the teacher meets w ith students one on one or in small groups to check on progress and, most importantly, to have " instructional conversations." The topic ofthese conversations is stud ent work and progress over time relative to the day's learning objective. Teachers provide descriptive feedback to each student and document what was discussed and decided through st andardized note-t aking and record-keeping protocols. Sharing, critique , and debriefing. At the end of each workshop, t eachers and stud ents take a few minutes to share progress, examine student work as small or large groups, or debrief the group's performance during that workshop . This immediate feedba ck is one Page 36 The Children's Guild, Ltd., DC Public Charter School ( • of several ways in which students receive substantive feedback on their progress from their peers and teachers . Assessment. Multiple form s of informal and forma l assessment of student progress are part of every workshop. Mini-lessons, conferences, and the daily debriefing all give the teacher opport unities to assess student understanding. Teachers then use what they have learned about student understanding to give students feedback, plan for reteaching, and provide immediate interventions for struggling students. Some workshops end with students writing or respond ing orally with an "exit ticket" that gives the teacher more information about individual students' progress on the daily learning target. 3. A lesson design that always uses learning ta rgets, but may vary in lesson format {depending on the focus of the learning target), bu ilds student engagement, sets a clear purpose and vision for quality and concludes by helping students synthesize current understanding of the content and skills and reflect on their progress toward learning target. Strategies used in lesson design maximize engagement and deep learning by doing the following: ( Strategically select and sequence instructional practices within and across lessons. Strategically use learning targets and knowledge of their students to plan lessons. Strategically se lect a lesson format {e.g. workshop, discovery-based, protocol-based, lecture, video, work sessions, labs and games) • Craft each lesson that begins by building student engagement and setting clear purpose. Teachers address the fo llowing questions when planning: 1. How will this lesson or series of lessons help students make progress toward the learning target{s)? 2. What will cause students to be cu ri ous and want to learn? 3. How will I provide students with a visio n of the learning target{s) in a way that gives them ownership of their learning? • Structure lessons so that they talk less and students talk more; they set up students to do the thinking and the work, they scaffold instruction in the body of lessons to ensure student success by addressing the following questions: 1. What sequenced steps wi ll the students and I take to ensure that all students meet the learning target{s)? 2. How will students know what quality looks like, and how will I support them in producing quality work? 3. How wi ll students work or practice together during learning? • Conclude lessons by helping students synthesize their current understanding of the content and skills focused on in the lesson and reflect on the ir progress toward the learning targets. The teacher uses information gleaned from the students' synthesis to plan subsequent lessons. Teachers address the following questions when planning: 1. How will my students demonstrate and/or synthes ize their understanding? 2. How will I use this information to plan my next instructional steps? • • • ( The Children's Guild, Ltd., DC Pub lic Charter School Page 37 ( • Sometimes teachers may start a lesson or an investigat ion with a complex or provocative prob lem and will build students' skills, vocabulary, and concepts on a " need to know" basis; alternatively, teachers will sometimes start a lesson or an investigation with an experience, and invite students to make sen se of it. Actively engage and guide st udents (for examp le, confer with students, pull student s into small invitational groups, etc.) during st udents' ind ependent work times. Teachers embed differentiation strategies within lessons to ensure that all students are effect ively supported and appropriate ly cha llenged. ( 4. A balance of whole group and small group experiences that emphasize collaboration, choice and shared respon sibility. Examples of these strategies are as follows: • Introduction . The introduction t aps into students' curiosity, sets a positive tone, builds the need to know, and links to previous learning. Th e learning target is shared during the introduction. • Protocols. Teachers will use protocols (such as Socratic seminars, learning logs, and j igsaws) to en sure that all students th ink crit ically and participate fu lly. In addition, they will use protocols to look at student work (for examp le, Coll aborative Assessment Conference), faci litate class room meetings and student advisory periods, and model and encourage behavior conducive to productive individual and group work. • Workshops. Teachers will use the workshop format to model or demonstrate a concept, skill, or strategy; require st udents to practice and apply what was modeled; and discuss and debrief what has been learned . • M ini-lessons. Teachers will use mini-lessons to introduce and explicitly teach concepts, skills, and strategies to th e whole class or small groups, as needed, often in response to student work and misconceptions. • Modeling. Teachers will use demonstrations, role-plays, and fishbowls to set criteria and model expectations for high quality group process, products, writing, reading, and problem-solving. Teachers w ill also use think-a louds to model comprehension strategies and skills . • Representative thinking. Teachers will use anchor charts and other forms of documentation to synthesize and make student understanding public. Students wi ll represent their thinking usin g formats such as graphic organizers, recording form s, journals, quick-writes, and summaries ofth eir learning. • Questioning and following st udent thinking. Teach ers will ask open-ended questions and follow-up questions to stimulate student thinking. Teachers will confer with students individually and in small groups on a regular basis to gauge each student's level of understa nding, differentiate instruction, and identify issues affecting a whole class. • Using exemplars and models. Teachers w ill use a range of exemp lars and models to help students see and understand quality, format, and group work. Teache rs w ill use exemplars and models to elicit criteria and construct rubrics. The Chi ldren's Guild, Ltd., DC Public Charter School Page 38 ( â€˘ â€˘ Multiple drafts, revision, and critique. Students will produce multiple drafts for all products and assess each draft against generated criteria and rubrics to improve successive drafts. Teachers will develop focused questions to guide revision . Students will use critique protocols to receive and provide feedback and to revise their work. Reflecting and debriefing. Teachers and students will use reflection and debriefing of lessons and experiences to improve retention of information, generalization, and transfer of learning, and to set goals for future learning. 5. Fieldwork and outside experts as learning resources. Using natural and social environments of the school's commun ity as sites for purposeful fieldwork and using professional experts and citizens with firsthand knowledge of events and issues will support accuracy, integrity and quality in students' work . Field work sites are used for purposeful connections to academic work. Students working in the field will be active investigators using the research tools, techniques of inquiry, and standards of presentation used by professionals in the field. CGDC will develop procedures and protocols to ensure that fieldwork is safe and productive. In addition to having students conduct research outside the schoo l, teachers will also bring experts from the community into the classroom . Older students participate in internships and apprenticeships that engage them in the real world . ( 6. Arts Integration - In CGDC, art in all forms will be celebrated as a foundat ion of culture and a central aspect of learning and life. Artistic skills will be understood as intelligences, and artistic achievement is valued as academic achievement. CGDC student's Exhibitions of Learning will feature the arts along with other subjects. Student art work will fill the school and will be displayed in a way that honors the work. The visual and performing arts will be taught using the same effective instructional practices that are used in other disciplines and all students will have access to professional artists and professional exh ibitions and performances. CGDC understands that the arts build school culture and student character by emphasizing authentic performance, craftsmanship, risk-taking, creativity, and a quest for beauty and meaning. The heritage of critique in the arts will form the basis for a whole-school culture of critique in all disciplines. Information and Communication Technologies (ICT} Integration CGDC recognizes the ever-increasing impact of information and communication technologies (ICT) on teaching and learning. All CGDC staff will be trained to use the technologies provided and use these technologies to enhance and support learning and teaching. ICT provides opportunities for the enhancement of learning, and may significantly support students in their inquiries, and in developing their conceptual understand ing. CGDC views technology as a tool for learning, albeit with its own set of skills, as opposed to an additional subject area. ICT ski lls should be developed and learned in order to support the needs of individual learners in their inquiries. 7. The Children's Guild, Ltd ., DC Public Charter School Page 39 ( The use of ICT can : • Document the learning, making it available to all parties • Provide opportun it ies for rapid feedback and reflection • Provide opportunities to enhance authentic learning • Provide access to a broad range of sources of information • Provide stude nts with a range of tools to store, organ ize and present their learning • Facilitate communication with a wide-ranging audience Various forms of technology will be ava ilable for students at CGDC. Classrooms wil l be equipped with a SmartBoard, netbooks, computer stations and various software programs to support their curriculum. CGDC will pi lot an iPad program to enhance instruction with special education students. The Children's Guild has been effectively implementing an iPad program w ith their autism population using APPS geared towards reading, math, writing, communication, social stories and creative expression. Developing digital citizenship among students at CGDC will enable students to competently navigate the internet and digital world . Acce leration and remediation software programs wil l enhance t he curriculum made avai lable to al l students within the school community. These methods are app licable to all grade levels and subject areas. They are the structural bu ilding blocks that allow CGDC teachers to differentiate, organ ize t ime space, materials and students to best meet the every student's learning needs and create engaging experiential and problem-based learning as a model for life long-learn ing. They allow teachers to talk less and students to talk more and do more thinking within the classroom, engage students in the curiosity of learn ing, and ask them to take more ownership of their own learning. They support students in embracing struggle as essential to growth and transformation. Students learn to respect and value a positive relationship with nature and divergent ideas. They enable students to use a variety of manipulatives as tools for thinking and representing and taking responsibility for producing something that shows their individual thinking. CGDC's methods of instruction work in concert and support one another in promoting high achievement through active learning, character growth and teamwork. These methods are the framework in which the school's goals are addressed. TEACCH methods will be used throughout the school and in particular w ith students with disabilities. Four essential components of the TEACCH philosophy will be utilized to help students access the curriculum in meaningfu l ways. The arrangement of the physical environment is important for increasing understanding by reducing anxiety which directly impact's the student's behavior and ability to learn. A clear, consistent and predictable physical environment enables students to develop the organizational skills needed to access the curriculum across subjects. The second essential component is the use of visual schedules. These visual schedules help inform students of what occurs in their day and in what sequence events will happen. By reducing confusion and increasing flexibility, students have great success with accessing their activities throughout their day. Schedu les also help with The Chi ldren's Guild, Ltd., DC Public Charter School Page 40 ( l ( transitions by preparing students for what comes next and eventually leading to independence. A work system is the third component of the TEACCH model. Individualized work systems help keep students organized so they can function independently and effectively in a variety of different activities . Students learn how to make choices and develop personal autonomy through individualized work systems. Visual structures are the fourth component of the TEACCH model. Visua l structures are used to differentiate tasks for students by highlighting, organizing and clarifying important and relevant information . Carefu lly assessing the visual structures needed by each student to access the curriculum w ill enable him/her to be successful. CGDC wi ll administer formative assessments to all students at least three t imes per grading period using assessments aligned to the cess and to learning targets in reading and math in order to accommodate the different learning styles anq needs of all students. For students who fai l to reach specified cut scores on the formative benchmark assessment or whom teachers refer because they are in danger of failing, teachers will construct a more extensive Student Learning Profile. Th is profile triangu lates all available data- including PARCC data, benchmark data, and individual assessment data to draw conclusions regarding individual students' strengths and weaknesses and to place students into a specific intervention with progressive increases or decreases of support as needed. Gifted and talented students, along with other students in need of academic enrichment, are also identified through this process. These assessments and profi les w ill drive the efforts of the tiered acce lerated learning program. ( CGDC administrators will review behavioral data on an ongoing basis to determine students who are at risk of needing additiona l support either academically, behaviorally or both. Behavioral interventions will be identified and implemented to assist students in being available for learning in the classrooms. Additional supports will be secured, if needed, that will enable the student to acquire the needed skills to self-regulate and independently participate with the curriculum. All interventions will be monitored to determine their effectiveness as well as modified when necessary. Upon enrollment information regarding English proficiency will be obtained. The non-English Proficient (NEP) or limited English proficient (LEP) student will be placed in a class appropriate to the child's academic needs as determined from the student's prior placement or by means of a home language survey and an assessment of English listen ing, speaking, reading, and writing skills that is considered reliable by DCPS. For the student who is NEP/LEP, a program to acquire or improve English language skills and cultural understanding will be made available to the student during the school day. Students who are NEP/LEP will be placed with age/grade peers. Ongoing consultation with the ESOL teacher will be instrumental in determining what kinds of ass istance each student will need to maximize instruction in the classroom and to ensure that the student is not misplaced or tracked inappropriately. Students identified as NEP/LEP will be evaluated annually in listening, speaking, reading, and writing English. NEP/LEP students will not be excluded from ( The Children' s Guild, Ltd ., DC Pub lic Charter School Page 41 ( any curricular or extra-curricular activity afforded to other students in the school. Any supports needed by the student to ensure full participation in these programs will be made available. ESOL teachers may work one-on-one or in small groups w ith ESOL students outside the classroom, or as an academic support person in the classroom itself. ESOL teachers help ESOL students develop t he English reading, writing, speaking and listening skills that they need in order to understand and master the academic cont ent in the CGDC curriculum. To track their progress in English, ESOL students are tested w it h the Idea Prof iciency Test (I PT) t wice a year. When ESOL students are ab le to function in English at their highest academic potential in the regular classroom, they become "consult" ESOL students for a year. ESOL teachers consult with the regu lar classroom teachers to make sure that the students are doing well in their academ ic work. After one successful year as a "consult" student, students may be released from the ESOL program. ESOL teachers stay in close communication with their students' teachers as well as their students' families. Frequent communication regarding the student's progress both w ith the classroom teacher as well as the student's parents allows effective monitoring of the learning process. Materials sent home will be in both English and the family's native language. Add itionally, translations w ill be made available if needed. ( ESOL teachers w ill collaborate and/or co-teach with the classroom teacher to ensure that NEP or LEP students are able to access the learning standards. ESOL teachers wi ll assist the classroom teacher in providing instructional strategies that assure NEP and LEP students are able to engage in classroom learning activities fully. Curriculum materials will be modified, or provided in a Spanish version, so NEP and LEP stude nts are provided with the needed curriculum resources. The Dean of Student and Family Life will serve as the ESOL liaison with OSSE and will coord inate with the OSSE ESOL office throughout the year to ensure that all required ESOL and related services are being provided . CGDC will implement a consistent open enrollment pol icy in accordance with the requi rements set forth by PCSB. CGDC will abide by all federa l, state, and local education requirements to serve all students including students with disabilities, LEP, and Gifted and Talented st udents. To complete the continuous instructional loop, teachers will meet regularly during common planning time to review student progress, examine student work, and share effective instructional strategies. This process is formally connected to embed professional development and involves a multi-disciplinary team including teachers, instructional support staff, special educators, and administrators. Students who are experiencing academic difficulties may also need increased supports. They may be experiencing challenges at home, health issues such as v ision and hearing, or personal and socia l issues that may become barriers to learn ing. Accelerated learners wi ll require modifications to instruction through advanced leveled text, The Chi ldren's Guild, Ltd., DC Public Charter Schoo l Page 42 ( ( differentiated higher order thinking questions, electives and research topics that allow them to delve into topics in a deeper fashion . CGDC anticipates the student population in Ward 7 will be similar to populations served in our existing charter schools in addition to our nonpublic specia l education schools. This population will require targeted interventions to engage the students in rigorous curriculum and authentic learning experiences. CGDC's methods of instruction are derived from the mission, beliefs and va lues of the school and provide students with college and career ready skills. These goals include student engagement in the learning process, development of new skills, prevention of behavior problems through proactive intervention strategies, development of critical thinking skills, the ability to problem solve creatively, development of self discipline and a commitment to serve a cause larger than oneself. Evidence of achieving these goals is in the student' s ability to demonstrate both academic and behavioral progress. CGDC's teaching strategies are the veh icle by wh ich the teaching goals and school outcomes become actual ized. The Common Core State Standards are the curriculum or "what" that instruction is based upon and Project Based Learning (PBL) is "how" students become involved w ith the curriculum. PBL is based upon the inquiry process and incorporates Bloom's taxonomy higher level skills: create, evaluate, analyze and apply. Centered on a real world problem, PBL incorporates creative problem solving, researching potential solutions, interfacing with various forms of technology and the production of a product that demonstrates their learning. The PBL products/projects provide opportunities for art integration as children integrate cross curricu lar concepts into a finished product that may include a performance, music, art or technology. The TEACCH principles of visual structures, schedules, work systems and physical organization provide the foundation for answering the 5 basic questions students have about the learning experience: ( 1. 2. 3. 4. 5. Where am I going? What am I going to do when I get there? How much do I have to do? When will I be finished? What will I do next? ( Neuroscience has shown that when there is an absence of fear and students understand what is expected of them, they increase their cognitive avai lability for learning. As a result, students can function independently within the school environment when they have a clear understanding of what is expected of them. TEACCH also emphasizes the importance of incorporating the students' strengths when teach ing new skills. Teachers integrate the students' strengths when they make real work connections and help students apply what they are learning in the classroom with their community. Technology is infused t hroughout the instructional process allowing students to understand the global economy in addition to providing an accommodation for students with disabilities. The Children's Guild, Ltd., DC Public Charter School Page 43 ( '----- d. Strategies for Providing Intensive Academic Support Academic Support Teachers will implement differentiated instructional practices to meet the needs of students who are substantially below grade level in reading and math . Instructional materials and resources are differentiated to meet the lexile ranges (MAP) and level readers (Fountas and Pinnell) of the students including articles, novels, journals, close reading, internet articles, websites, remedial reading programs and educational software. The use of manipulatives in mathematics provide students with concrete examples of mathematical concepts and hands-on learning experiences. Remedial math programs such as Fast Math, Underst anding Math, and Envision Math Diagnosis and Intervention System also provide additional instructional support for students performing significantly below grade level. Instructional coaches are available to work with classroom teachers to determine effective interventions aligned with the learning profiles of the student. Tier 1 is regular classroom instruction, differentiated as much as possible within the classroom to meet the individual students' needs. We anticipate approximately 80% of the students will succeed in this tier as they are achieving the learning targets prescribed in the cess at the proficient or advanced level. The core reading and math curriculum is a high quality instructional program that is implemented in alignment with the cess. Tier 2 instruction is characterized by small group instruction (composed of three to six students), meeting three to four times each week for 30 to 60 minutes each, for nine to twelve weeks. We anticipate that app roximately 15% of the students will be served by Tier 2 interventions. The needs of these students are identified through the assessment process and in struct ional interventions are delivered from the focus of the specific targeted needs. Remediation of the targeted skill is provided in the small group setting using supplemental interventions identified fo r students at ri sk of not meeting academic standards. Tier 3 instruction is characterized by one-to-one or small group instruction (for one to three students), meeting daily for 30 to 60 minutes each, for nine to 12 weeks. We anticipate that Tier 3 students will be approximately 5% of the student population. Tier 3 students are considered to be at high risk for failure and if they are not responsive to an inten sive, specific intervention may be candidates for identification as having special education need s. Supplemental interventions for students at high risk are provided at this level. ( ( The Children's Guild, Ltd ., DC Public Charter School Page 44 ( ( Ongoing progress monitoring will occur to determine the effectiveness of the intervention and assess student progress. Students will receive tiered interventions until they are able to successfully access the general curriculum without additional support or supplemental materials. Tiered interventions can be accessed throughout the school year on an as needed basis to support the student' s learning style and needs as they encounter new skills in the curriculum. Continuum of Services CGDC has developed an operations model designed to provide holistic and seamless service delivery for students with disabilities in collaboration with The Children's Guild, Inc. The model centers on the special education teacher(s) and the IEP coordinator as the case managers directly responsible and accountable for en suring the academic progress, individual case compliance, and the Free Appropriate Public Education {FAPE) requirement. CGDC will provide a continuum of special education services in order to meet the intended needs of the identified special education population . CGDC proposes to have a large special education population {60%) with students at all levels of service and will align the special education staffing caseload to match the service needs of its students. ( The Children's Guild, Ltd ., DC Public Charter School Page 45 at the 7C Advisory Neighborhood Commission meeting. Our plan for continued engagement with Ward 7 is as follows: Meet with Greg Stewart and Walter Garcia of the 7C Advisory Neighborhood Commission and attend the meeting held at Sargent Memorial Presbyterian Church, located at 5109 Nannie Helen Burroughs Ave., NE on March 13th at 7 p.m. Attend Meeting of ANC 7B at Ryland Methodist Church, 3200 S. Street SE, March 20th at 7 p.m. Attend Meeting of ANC 7D at Sixth District Police Station, 100 42nd St. NE, March 10th at 6:30 p.m. Attend Meeting of ANC 7E at Jones Memorial Church, 4725 G St. SE, March 10th or April 14, 7 p.m. Attend Meeting of ANC 7F at Washington Tennis and Education Foundation, 200 Stoddert Place, March 18 at 6:30 p.m. We also are in the process of scheduling meetings with Council Members Kenyon McDuffie ’s and Marion Berry's staff given the number of special needs students in Wards 5 and 8 and the proximity of the school to their constituents. We have received four letters of support, i.e., one from the See Forever Foundation, two from parents, and one from DCASE. See Appendix, Section R. Community Partnerships The CGDC educational model emphasizes Project Based Learning which involves students in expeditions into the community where field work, research, exploration and access to experts creates an authentic, real-world learning experience. Thus, CGDC will seek to create strong business partnerships in every grade level that support these learning expeditions. In particular, Washington, DC offers excellent opportunities to form business partnerships in the healthcare, research, hospitality and transportation industries. The Smithsonian and other museums also provide the opportunity to create partnerships that will create memorable learning expeditions. In addition to Project-Based Learning partnerships, CGDC also emphasizes Arts Integration at every grade level and partnerships with arts organizations within the District will be a primary goal of the school. Character Development is another important aspect of the CGDC model and service learning opportunities are an important tool in teaching students how to give back to their community. The District is home to many charity headquarters where such opportunities can be explored. The District also has many skilled community change agents who can teach students how to positively impact their own neighborhoods and communities through service. Developing these opportunities would also become a significant goal of school leadership. In addition, CGDC has collaborated with the See Forever Foundation/Maya Angelou Charter School regarding its school building and has reached out to the DC Special Education Cooperative regarding special education services in charter schools. The Children’s Guild, Ltd., DC Public Charter School Page 75 New York State Common Core 3 GRADE Mathematics Curriculum GRADE 3 • MODULE 4 Table of Contents GRADE 3 • MODULE 4 Multiplication and Area Module Overview ......................................................................................................... i Topic A: Foundations for Understanding Area ...................................................... 4.A.1 Topic B: Concepts of Area Measurement ............................................................... 4.B.1 Topic C: Arithmetic Properties Using Area Models................................................. 4.C.1 Topic D: Applications of Area Using Side Lengths of Figures ..................................4.D.1 Module Assessments ............................................................................................. 4.S.1 Module 4: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Multiplication and Area 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. i NYS COMMON CORE MATHEMATICS CURRICULUM 3 Module Overview Lesson New York State Common Core Grade 3 â€˘ Module 4 Multiplication and Area OVERVIEW In this 20-day module students explore area as an attribute of two-dimensional figures and relate it to their prior understandings of multiplication. In Grade 2, students partitioned a rectangle into rows and columns of same-sized squares and found the total number by both counting and adding equal addends represented by the rows or columns (2.G.2, 2.OA.4). In Topic A, students begin to conceptualize area as the amount of twodimensional surface that is contained within a plane figure. They come to understand that the space can be tiled with unit squares without gaps or overlaps (3.MD.5). They make predictions and explore which rectangles cover the most area when the side lengths differ (but area is actually the same). Students may, for example, cut and fold rectangles to confirm predictions about whether a 1 by 12 rectangle covers more area than a 3 by 4 or a 2 by 6 rectangle. They reinforce their ideas by using inch and centimeter square manipulatives to tile the same rectangles and prove the areas are equal. Topic A provides studentsâ€™ first experience with tiling, from which they learn to distinguish between length and area by placing a ruler with the same size units (inches or centimeters) next to a tiled array to discover that the number of tiles along a side corresponds to the length of the side (3.MD.6). In Topic B, students progress from using square tile manipulatives to drawing their own area models. Anticipating the final structure of an array, they complete rows and columns in figures such as the example shown at the right. Students connect their extensive work with rectangular arrays and multiplication to eventually discover the area formula for a rectangle, which is formally introduced in Grade 4 (3.MD.7a). In Topic C, students manipulate rectangular arrays to concretely demonstrate the arithmetic properties in anticipation of the following lessons. They do this by cutting rectangular grids and rearranging the parts into new wholes using the properties to validate that area stays the same, despite the new dimensions. They apply tiling and multiplication skills to determine all whole number possibilities for the side lengths of rectangles given their areas (3.MD.7b). 8 cm Topic D creates an opportunity for students to solve problems involving area (3.MD.7b). Students decompose and/or compose composite regions like the one shown at right into non-overlapping rectangles, find the area of each region, and add or subtract to determine the total area of the original shape. This leads students to design a simple floor plan that conforms to given area specifications (3.MD.7d). 2 cm 2 cm 5 cm 1 by 12 3 by 4 2 by 6 Module 4: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Multiplication and Area 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. ii NYS COMMON CORE MATHEMATICS CURRICULUM 3 Module Overview Lesson New York State Common Core Focus Grade Level Standards Geometric Measurement: understand concepts of area and relate area to multiplication and to addition. 3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement: a. b. 3.MD.6 3.MD.7 A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). Relate area to the operations of multiplication and addition. a. b. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning. Module 4: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Multiplication and Area 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. iii NYS COMMON CORE MATHEMATICS CURRICULUM 3 Module Overview Lesson New York State Common Core c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a Ă— b and a Ă— c. Use area models to represent the distributive property in mathematical reasoning. Recognize area as additive. Find the areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems. d. Foundational Standards 2.MD.1 2.MD.2 2.G.2 Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes. Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. Focus Standards for Mathematical Practice MP.2 Reason abstractly and quantitatively. Students build toward abstraction starting with tiling a rectangle, then gradually moving to finishing incomplete grids and drawing grids of their own, then eventually working purely in the abstract, imaging the grid as needed. Construct viable arguments and critique the reasoning of others. Students explore their conjectures about area by cutting to decompose rectangles and then recomposing them in different ways to determine if different rectangles have the same area. When solving area problems, students learn to justify their reasoning and determine whether they have found all possible solutions, when multiple solutions are possible. Attend to precision. Students precisely label models and interpret them, recognizing that the unit impacts the amount of space a particular model represents, even though pictures may appear to show equal sized models. They understand why when side lengths are multiplied the result is given in square units. Look for and make use of structure. Students relate previous knowledge of the commutative and distributive properties to area models. They build from spatial structuring to understanding the number of area-units as the product of number of units in a row and number of rows. Look for and express regularity in repeated reasoning. Students use increasingly sophisticated strategies to determine area over the course of the module. As they analyze and compare strategies, they eventually realize that area can be found by multiplying the number in each row by the number of rows. MP.3 MP.6 MP.7 MP.8 Module 4: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Multiplication and Area 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. iv NYS COMMON CORE MATHEMATICS CURRICULUM 3 Module Overview Lesson New York State Common Core Overview of Module Topics and Lesson Objectives Standards Topics and Objectives 3.MD.5 3.MD.6 3.MD.7 A Foundations for Understanding Area Lesson 1: Understand area as an attribute of plane figures. Lesson 2: Lesson 3: Lesson 4: 3.MD.5 3.MD.6 3.MD.7a 3.MD.7b 3.MD.7d B Decompose and recompose shapes to compare areas. Model tiling with centimeter and inch unit squares as a strategy to measure area. Relate side lengths with the number of tiles on a side. 4 Days 4 Concepts of Area Measurement Lesson 5: Form rectangles by tiling with unit squares to make arrays. Lesson 6: Lesson 7: Lesson 8: Draw rows and columns to determine the area of a rectangle, given an incomplete array. Interpret area models to form rectangular arrays. Find the area of a rectangle through multiplication of the side lengths. Mid-Module Assessment: Topics A–B (assessment 1 day, return ½ day, remediation or further applications ½ day) 3.MD.5 3.MD.6 3.MD.7a 3.MD.7b 3.MD.7c 3.MD.7d C Arithmetic Properties Using Area Models Lesson 9: Analyze different rectangles and reason about their area. Lesson 10: Lesson 11: Apply the distributive property as a strategy to find the total area of a large rectangle by adding two products. Demonstrate the possible whole number side lengths of rectangles with areas of 24, 36, 48, or 72 square units using the associative property. 2 3 3.MD.6 3.MD.7a 3.MD.7b 3.MD.7c 3.MD.7d 3.MD.5 D Applications of Area Using Side Lengths of Figures Lesson 12: Solve word problems involving area. Lessons 13–14: Find areas by decomposing into rectangles or completing composite figures to form rectangles. Lessons 15–16: Apply knowledge of area to determine areas of rooms in a given floor plan. End-of-Module Assessment: Topics A–D (assessment 1 day, return ½ day, remediation or further applications ½ day) 5 2 20 Total Number of Instructional Days Module 4: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Multiplication and Area 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. v NYS COMMON CORE MATHEMATICS CURRICULUM 3 Module Overview Lesson New York State Common Core Terminology New or Recently Introduced Terms Area (the amount of two-dimensional space in a bounded region) Area model (a model for multiplication that relates rectangular arrays to area) Module 1 and Module 3 Module 4 Square unit (a unit of area—specifically square centimeters, inches, feet, and meters) Tile (to cover a region without gaps or overlaps) Unit square (e.g., given a length unit, it is a 1 unit by 1 unit square) Whole number (an integer, a number without fractions) Familiar Terms and Symbols1 Array (a set of numbers or objects that follow a specific pattern, a matrix) Commutative Property (e.g., rotate a rectangular array 90 degrees to demonstrate that factors in a multiplication sentence can switch places) Distribute (e.g., 2 × (3 + 4) = 2 × 3 + 2 × 4) Geometric shape (a two-dimensional object with a specific outline or form) Length (the straight-line distance between two points) Multiplication (e.g., 5 × 3 =15) Rows and columns (e.g., in reference to rectangular arrays) Suggested Tools and Representations Area model Array Grid paper (inch and centimeter) Rulers (both centimeter and inch measurements) Unit squares in both inch and centimeter lengths (e.g., square tiles used for measuring area) 1 These are terms and symbols students have seen previously. Module 4: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Multiplication and Area 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. vi NYS COMMON CORE MATHEMATICS CURRICULUM 3 Module Overview Lesson New York State Common Core Scaffolds2 The scaffolds integrated into A Story of Units give alternatives for how students access information as well as express and demonstrate their learning. Strategically placed margin notes are provided within each lesson elaborating on the use of specific scaffolds at applicable times. They address many needs presented by English language learners, students with disabilities, students performing above grade level, and students performing below grade level. Many of the suggestions are organized by Universal Design for Learning (UDL) principles and are applicable to more than one population. To read more about the approach to differentiated instruction in A Story of Units, please refer to “How to Implement A Story of Units.” Assessment Summary Type Mid-Module Assessment Task End-of-Module Assessment Task Administered After Topic B Format Constructed response with rubric Standards Addressed 3.MD.5 3.MD.6 3.MD.7abd 3.MD.5 3.MD.6 3.MD.7a–d After Topic D Constructed response with rubric 2 Students with disabilities may require Braille, large print, audio, or special digital files. Please visit the website, www.p12.nysed.gov/specialed/aim, for specific information on how to obtain student materials that satisfy the National Instructional Materials Accessibility Standard (NIMAS) format. Module 4: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Multiplication and Area 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. vii New York State Common Core GRADE 3 Mathematics Curriculum GRADE 3 • MODULE 4 Topic A Foundations for Understanding Area 3.MD.5, 3.MD.6, 3.MD.7 Focus Standard: 3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement: a. b. Instructional Days: Coherence -Links from: 4 G2–M2 G3–M1 G3–M3 -Links to: G4–M3 G4–M7 Addition and Subtraction of Length Units Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 Multiplication and Division with Units of 0, 1, 6–9, and Multiples of 10 Multi-Digit Multiplication and Division Exploring Multiplication A square with side length 1 unit, called a “square unit,” is said to have “one square unit” of area, and can be used to measure area. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. In Lesson 1, students come to understand area as an attribute of plane figures that is defined by the amount of two-dimensional space within a bounded region. Students use pattern blocks to tile given polygons without gaps or overlaps, and without exceeding the boundaries of the shape. Lesson 2 takes students into an exploration in which they cut apart paper rectangles into same-sized squares to concretely define a square unit, specifically square inches and centimeters. They use these units to make rectangular arrays that have the same area, but different side lengths. Lessons 3 and 4 introduce students to the strategy of finding area using centimeter and inch tiles. Students use tiles to determine the area of a rectangle by tiling the region without gaps or overlaps. They then bring the ruler (with corresponding units) alongside the array to discover that the side length is equal to the number of tiles required to cover one side of the rectangle. From this experience, students begin to relate total area with multiplication of side lengths. Topic A: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Foundations for Understanding Area 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported.License. 4.A.1 NYS COMMON CORE MATHEMATICS CURRICULUM Topic A 3â€˘ 4 A Teaching Sequence Towards Mastery of Foundations for Understanding Area Objective 1: Understand area as an attribute of plane figures. (Lesson 1) Objective 2: Decompose and recompose shapes to compare areas. (Lesson 2) Objective 3: Model tiling with centimeter and inch unit squares as a strategy to measure area. (Lesson 3) Objective 4: Relate side lengths with the number of tiles on a side. (Lesson 4) Topic A: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Foundations for Understanding Area 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported.License. 4.A.2 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 3• 4 Lesson 1 Objective: Understand area as an attribute of plane figures. Suggested Lesson Structure Fluency Practice Application Problem Concept Development Student Debrief Total Time (15 minutes) (5 minutes) (30 minutes) (10 minutes) (60 minutes) Fluency Practice (15 minutes) Group Counting 3.OA.1 Identify the Shape 2.G.1 Find the Common Products 3.OA.7 (4 minutes) (3 minutes) (8 minutes) Group Counting (4 minutes) Note: Group counting reviews interpreting multiplication as repeated addition. Direct students to count forward and backward, occasionally changing the direction of the count. Threes to 30 Sixes to 60 Sevens to 70 Eights to 80 Nines to 90 Identify the Shape (3 minutes) Materials: (T) Images of polygons (S) Personal white boards Note: This fluency reviews properties and vocabulary that will be used during today’s Concept Development. T: S: T: S: (Project a triangle.) How many sides does this shape have? 3. Name the shape. Triangle. Lesson 1: Date: Understand area as an attribute of plane figures. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.3 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 3• 4 Continue with the following possible sequence: quadrilateral (trapezoid), quadrilateral (rhombus), quadrilateral (square), and quadrilateral (rectangle). Find the Common Products (8 minutes) Materials: (S) Blank paper Note: This fluency reviews multiplication patterns from G3–Module 3. T: T: T: T: S: T: Fold your paper in half vertically. On the left half, count by twos to 20 down the side of your paper. On the right half, count by fours to 40 down the side of your paper. Draw lines to match multiples that appear in both columns. (Match 4, 8, 12, 16, and 20.) (Write × 2 = 4, × 2 = 8, etc., next to each corresponding product on the left half of the paper.) Write the complete equations next to their products. (Write equations and complete unknowns.) (Write 4 = × 4, 8 = × 4, etc., next to each corresponding product on the right half of the paper.) Write the complete equations next to their products. (Write equations.) (Write 2 × 2 = × 4.) Say the equation including all factors. 2 × 2 = 1 × 4. (Write 2 × 2 = 1 × 4.) Write the remaining equal facts as equations. (Write 4 × 2 = 2 × 4, 6 × 2 = 3 × 4, 8 × 2 = 4 × 4, and 10 × 2 = 5 × 4.) What patterns do you notice in your equations? Each multiple of 4 is also a multiple of 2. S: T: S: T: S: T: S: T: S: Application Problem (5 minutes) Eric makes a shape with 8 trapezoid pattern blocks. Brock makes the same shape using triangle pattern blocks. It takes 3 triangles to make 1 trapezoid. How many triangle pattern blocks does Brock use? Note: This problem reviews the G3–Module 3 concept of multiplying using units of 8. Lesson 1: Date: Understand area as an attribute of plane figures. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.4 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 3• 4 Concept Development (30 minutes) Materials: (S) Pattern blocks, Problem Set Part 1: Using pattern blocks to understand area. T: NOTES ON MULTIPLE MEANS OF ACTION AND EXPRESSION: S: T: S: T: S: T: S: T: Look at Problem 1 on your Problem Set. Discuss with a partner whether you think Shape A or Shape B takes up more space. Be prepared to explain your answer. Manipulating pattern blocks may be a (After students discuss, facilitate a whole class challenge for some learners. Try the discussion.) following tips: Shape A, because it’s longer than Shape B. Shape B, Partner students so they can work because it’s taller than Shape A. together to cover the shapes. Use green triangle pattern blocks to cover Shape A and Encourage students to hold the pattern blocks in place with one Shape B. Be sure the triangles do not have gaps hand, while they place the remaining between them, they don’t overlap, and they don’t go blocks. outside the sides of the shapes. (Allow time for Instead of using pattern blocks, students to work.) What did you notice about the provide paper shapes that can be number of green triangles it takes to cover Shape A glued, so they won’t move around and Shape B? unnecessarily. It takes 6 green triangles to cover each shape! Offer the computer as a resource to create and move shapes. Answer Problem 1 on your Problem Set. (Allow time for students to write answers.) Do all the green triangles take up the same amount of space? Yes, because they’re all the same size. What does that mean about the amount of space Shape A and Shape B take up? They’re the same. It took 6 triangles to cover each shape, which means the shapes take up the same amount of space. The amount of space that Shape A takes up is equal to the amount of space Shape B takes up. The amount of flat space a shape takes up is called its area. Since Shapes A and B take up the same amount of space, their areas are equal. Repeat the process of using pattern blocks to cover Shapes A and B with the blue rhombus and the red trapezoid pattern blocks. Students record their work on Problems 2 and 3 in the Problem Set. T: S: What is the relationship between the size of the pattern blocks and the number of pattern blocks it takes to cover Shapes A and B? The bigger the pattern block, the smaller the number of pattern blocks it takes to cover these shapes. The bigger pattern blocks, like the trapezoid, cover more area than the triangles. That means it takes fewer trapezoids to cover the same area as the triangles. Answer Problem 4 on your Problem Set. T: Lesson 1: Date: Understand area as an attribute of plane figures. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.5 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 3• 4 Part 2: Measuring area using square units. T: Use orange square pattern blocks to cover the NOTES ON rectangle in Problem 5. Be sure the squares don’t have MULTIPLE MEANS gaps between them, they don’t overlap, and they don’t go outside the sides of the rectangle. (Allow students OF ENGAGEMENT: time to work.) How many squares did it take to cover Students working above grade level the rectangle? can be encouraged to find other square units in the classroom that they can 6! either use to make rectangles or that Answer Problem 5 on your Problem Set. (Allow time already form rectangles. Such items for students to write answers.) The area of Shape C is might include sticky notes, desktops, 6 square units. Why do you think we call them square floor tiles, and linking cubes. Students units? can create a poster to share with the class that shows the areas of the Because they’re squares! The units used to rectangles made with these other measure are squares, so they’re square units! square units. Yes! The units used to measure the area of the rectangle are squares. Use red trapezoid pattern blocks to cover the rectangle in Problem 5. Be sure the trapezoids don’t have gaps between them, they don’t overlap, and they don’t go outside the sides of the rectangle. (Allow students time to work.) What did you notice? It’s not possible! The red trapezoids can’t cover this shape without having gaps. Use this information to help you answer Problem 6 on your Problem Set. (Allow time for students to write answers.) I’m going to say an area in square units, and you’re going to make a rectangle with your pattern blocks that has that area. Which pattern blocks will you use? The squares because the units for area that you’re telling us are square units! Here we go! Four square units. (Make rectangles.) S: T: S: T: T: S: T: S: T: S: Continue with the following possible suggestions: 12 square units, 9 square units, and 8 square units. Invite students to compare their rectangles to a partner’s rectangles. How are they the same? How are they different? If time allows, students can work with a partner to create rectangles that have the same areas, but look different. Student Debrief (10 minutes) Lesson Objective: Understand area as an attribute of plane figures. The Student Debrief is intended to invite reflection and active processing of the total lesson experience. Invite students to review their solutions for the Problem Set. They should check work by comparing answers with a partner before going over answers as a class. Look for misconceptions or misunderstandings that can be addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the lesson. Lesson 1: Date: Understand area as an attribute of plane figures. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.6 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 3• 4 You may choose to use any combination of the questions below to lead the discussion. Talk to a partner. Do you think you can use orange square pattern blocks to cover Shapes A and B in Problem 1? Explain your answer. How many green triangle pattern blocks does it take to cover a blue rhombus pattern block? Use that information to say a division fact that relates the number of triangles it takes to cover Shape A to the number of rhombuses it takes to cover the same shape. (6 ÷ 2 = 3.) Explain to a partner how you used orange square pattern blocks to find the area of the rectangle in Problem 5. What new math vocabulary did we use today to communicate precisely about the amount of space a shape takes up? (Area.) Which units did we use to measure area? How did the Application Problem connect to today’s lesson? Exit Ticket (3 minutes) After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help you assess the students’ understanding of the concepts that were presented in the lesson today and plan more effectively for future lessons. You may read the questions aloud to the students. Lesson 1: Date: Understand area as an attribute of plane figures. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.7 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 Problem Set 3â€˘ 4 Name Date 1. Use green triangle pattern blocks to cover each shape below. Draw lines to show where the triangles meet. Then write how many triangle pattern blocks it takes to cover each shape. Shape A: _______ triangles Shape B: _______ triangles 2. Use blue rhombus pattern blocks to cover each shape below. Draw lines to show where the rhombuses meet. Then write how many rhombus pattern blocks it takes to cover each shape. Shape A: _______ rhombuses Shape B: _______ rhombuses 3. Use red trapezoid pattern blocks to cover each shape below. Draw lines to show where the trapezoids meet. Then write how many trapezoid pattern blocks it takes to cover each shape. Shape A: _______ trapezoids Shape B: _______ trapezoids Lesson 1: Date: Understand area as an attribute of plane figures. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.8 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 Problem Set 3â€˘ 4 4. How is the number of pattern blocks needed to cover the same shape related to the size of the pattern blocks? 5. Use orange square pattern blocks to cover the rectangle below. Draw lines to show where the squares meet. Then write how many square pattern blocks it takes to cover the rectangle. _______ squares 6. Use red trapezoid pattern blocks to cover the rectangle in Problem 5. Can you use red trapezoid pattern blocks to measure the area of this rectangle? Explain your answer. Lesson 1: Date: Understand area as an attribute of plane figures. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.9 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 Exit Ticket 3• 4 Name Date 1. Each is 1 square unit. Do both rectangles have the same area? Explain how you know. a. b. Lesson 1: Date: Understand area as an attribute of plane figures. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.10 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 Homework 3â€˘ 4 Name Date 1. Magnus covers the same shape with triangles, rhombuses, and trapezoids a. How many triangles will it take to cover the shape? _______ triangles b. How many rhombuses will it take to cover the shape? _______ rhombuses c. Magnus notices that 3 triangles from Part (a) cover 1 trapezoid. How many trapezoids will it take to cover the shape below? Explain your answer. _______ trapezoids Lesson 1: Date: Understand area as an attribute of plane figures. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.11 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 Homework 3â€˘ 4 2. Angela uses squares to find the area of a rectangle. Her work is shown below. a. How many squares did she use to cover the rectangle? _______ squares b. What is the area of the rectangle in square units? Explain how you found your answer. 3. Each is 1 square unit. Which rectangle has the biggest area? How do you know? Rectangle C Rectangle A Rectangle B Lesson 1: Date: Understand area as an attribute of plane figures. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.12 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 3• 4 Lesson 2 Objective: Decompose and recompose shapes to compare areas. Suggested Lesson Structure Fluency Practice Application Problem Concept Development Student Debrief Total Time (11 minutes) (5 minutes) (34 minutes) (10 minutes) (60 minutes) Fluency Practice (11 minutes) Group Counting 3.OA.1 Multiply by 4 3.OA.7 (4 minutes) (7 minutes) Group Counting (4 minutes) Note: Group counting reviews interpreting multiplication as repeated addition. Direct students to count forward and backward, occasionally changing the direction of the count. Sixes to 60 Sevens to 70 Eights to 80 Nines to 90 Multiply by 4 (7 minutes) Materials: (S) Multiply by 4 Pattern Sheet (6–10) Note: This activity builds fluency with multiplication facts using units of 4. It works toward students knowing from memory all products of two one-digit numbers. T: S: T: S: T: (Write 7 × 4.) Let’s skip-count up by fours. (Count with fingers to 7 as students count.) 4, 8, 12, 16, 20, 24, 28. What is 7 × 4? 28. Let’s see how we can skip-count down to find the answer, too. (Show 10 fingers.) Start at 10 fours, 40. (Count down with your fingers as students say numbers.) Lesson 2: Date: Decompose and recompose shapes to compare areas. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.13 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 3• 4 S: T: 40, 36, 32, 28. (Distribute Multiply by 4 Pattern Sheet.) Let’s practice multiplying by 4. Be sure to work left to right across the page. Distribute pattern sheet. Allow a maximum of two minutes for students to complete as many problems as possible. Direct students to work left to right across the page. Encourage skip-counting strategies to solve unknown facts. Continue with the following possible sequence: 9 × 4, 6 × 4, and 8 × 4. Directions for administration of Multiply By pattern sheet: 1. 2. 3. 4. Application Problem (5 minutes) Wilma and Freddie use patterns blocks to make shapes as shown. Freddie says his shape is bigger than Wilma’s because it’s longer than hers. Is he right? Explain your answer. Wilma’s Shape Freddie’s Shape Note: This problem reviews G3–M4–Lesson 1, specifically that even though shapes look different, they can have the same area. Concept Development (34 minutes) Materials: (S) Paper Strip 1: 1 in × 12 in, Paper Strip 2: 1 cm × 12 cm, scissors, ruler, Problem Set page 1 Students begin with Paper Strip 1. T: S: Measure your strip. How tall is it? 1 inch tall. Lesson 2: Date: Decompose and recompose shapes to compare areas. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.14 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 3• 4 T: Start at the edge of your strip and use your ruler to mark inches along the top. Do the same along the bottom. Use your ruler to connect the marks at the top to the matching marks at the bottom. NOTES ON MULTIPLE MEANS OF ACTION AND EXPRESSION: Make it easy for learners to mark inches and cut the strip with the following tips: Provide strips of thicker paper, such as cardstock. Provide strips of grid or graph paper to facilitate drawing lines. T: S: T: S: T: S: T: MP.6 S: T: S: T: S: T: T: S: T: T: S: T: S: T: How many units make up your strip? 12 units. Offer left-handed and adaptive scissors, if needed. What shape are they? They’re squares. Each one has 4 sides that are 1 inch. What is the area of the paper strip in square units? 12 square units! Since the sides of the squares each measure 1 inch, we call one of these squares a square inch. What is the area of your paper strip in square inches? 12 square inches! Did the number of squares change? No. Talk to a partner. What changed about the way we talked about the area of the paper strip? The units changed. Before we called them square units, but now we can call them square inches because all 4 sides measure 1 inch. We named this square unit. A square unit could have sides of any length. A square inch is always the same thing. Cut your paper strip along the lines you drew. Now rearrange all 12 squares into 2 equal rows. Remember, the squares have to touch but can’t overlap. Draw your rectangle in the chart for Problem 1. What is the area of the rectangle? 12 square inches. Record the area. You can record it by writing 12 square inches, or you can write 12 sq in. Rearrange all 12 squares into 3 equal rows to make a new rectangle. Draw it in the chart and record the area. At my signal, whisper the area of your rectangle to a partner. (Signal.) 12 square inches. Rearrange all 12 squares into 4 equal rows to make a new rectangle. Draw it in the chart and record the area. At my signal, whisper the area of your rectangle to a partner. (Signal.) 12 square inches. How is it possible that these three different rectangles and our paper strip all have the same area? If you offer paper strips with predrawn tick marks, guide discovery of inches. Darken lines for cutting. Lesson 2: Date: Decompose and recompose shapes to compare areas. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.15 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 3• 4 S: We used the same squares for each one, so they all have the same area. We rearranged 12 square inches each time. Just rearranging them doesn’t change the area. Repeat the process with Paper Strip 2 (1 cm × 12 cm). Note: The square inch and square centimeter tiles will be used again in G3–M4–Lesson 7. You may want to collect them or have students store them in a safe place. NOTES ON MULTIPLE MEANS OF ENGAGEMENT: Students working above grade level may enjoy more autonomy as they explore and compare area. Offer the choice of a partner game in which Partner A constructs a shape, after which Partner B constructs a shape with a greater or lesser area. Encourage students to modify the game or invent another that compares area. Problem Set (10 minutes) Students should do their personal best to complete the Problem Set within the allotted 10 minutes. Some problems do not specify a method for solving. This is an intentional reduction of scaffolding that invokes MP.5, Use Appropriate Tools Strategically. Students should solve these problems using the RDW approach used for Application Problems. For some classes, it may be appropriate to modify the assignment by specifying which problems students should work on first. With this option, let the careful sequencing of the Problem Set guide your selections so that problems continue to be scaffolded. Balance word problems with other problem types to ensure a range of practice. Assign incomplete problems for homework or at another time during the day. Student Debrief (10 minutes) Lesson Objective: Decompose and recompose shapes to compare areas. The Student Debrief is intended to invite reflection and active processing of the total lesson experience. Invite students to review their solutions for the Problem Set. They should check work by comparing answers with a partner before going over answers as a class. Look for misconceptions or misunderstandings that can be addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the lesson. You may choose to use any combination of the questions below to lead the discussion. Talk to a partner. What new units did we define today? Look at Problem 4. If Maggie uses square inches for Shape A and square centimeters for Shape B, which shape has a larger area? Lesson 2: Date: Decompose and recompose shapes to compare areas. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.16 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 3• 4 How do you know? Compare the shape you drew in Problem 5 to a partner’s. Are they the same? Do they have the same area? Why or why not? We started our lesson by using an inch ruler to break apart a rectangle into square inches. Turn and talk to a partner. Why was it important to break apart the rectangle into square inches? Exit Ticket (3 minutes) After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help you assess the students’ understanding of the concepts that were presented in the lesson today and plan more effectively for future lessons. You may read the questions aloud to the students. Lesson 2: Date: Decompose and recompose shapes to compare areas. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.17 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 Pattern Sheet 3• 4 Multiply. Lesson 2: Date: Decompose and recompose shapes to compare areas. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.18 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 Problem Set 3â€˘ 4 Name Date 1. Use all of Paper Strip 1, which you cut into 12 square inches, to complete the chart below. Drawing Area Rectangle A Rectangle B Rectangle C 2. Use all of Paper Strip 2, which you cut into 12 square centimeters, to complete the chart below. Drawing Area Rectangle A Rectangle B Rectangle C Lesson 2: Date: Decompose and recompose shapes to compare areas. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.19 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 Problem Set 3â€˘ 4 3. Compare the areas of the rectangles you made with Paper Strip 1 and Paper Strip 2. What changed? Why did it change? 4. Maggie uses her square inch pieces to create these two rectangles. Do the two rectangles have the same area? How do you know? Shape B Shape A 5. Count to find the area of the rectangle below. Then draw a different rectangle that has the same area. Lesson 2: Date: Decompose and recompose shapes to compare areas. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.20 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 Exit Ticket 3â€˘ 4 Name Date 1. Each is a square unit. Find the area of the rectangle below. Then draw a different rectangle with the same number of square units. 2. Zach creates a rectangle with an area of 6 square inches. Luke makes a rectangle with an area of 6 square centimeters. Do the two rectangles have the same area? Why or why not? Lesson 2: Date: Decompose and recompose shapes to compare areas. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.21 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 Homework 3â€˘ 4 Name Date 1. Each is a square unit. Count to find the area of each rectangle. Then circle all the rectangles with an area of 12 square units. a. b. c. Area = _______ square units Area = _______ square units Area = _______ square units f. d. e. Area = _______ square units Area = _______ square units Area = _______ square units Lesson 2: Date: Decompose and recompose shapes to compare areas. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.22 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 Homework 3â€˘ 4 2. Colin uses square inch pieces to create these rectangles. Do they have the same area? Explain. 3. Each is a square unit. Count to find the area of the rectangle below. Then draw a different rectangle that has the same area. Lesson 2: Date: Decompose and recompose shapes to compare areas. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.23 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 3 Lesson 3 Objective: Model tiling with centimeter and inch unit squares as a strategy to measure area. Suggested Lesson Structure Fluency Practice Application Problem Concept Development Student Debrief Total Time (13 minutes) (5 minutes) (32 minutes) (10 minutes) (60 minutes) Fluency Practice (13 minutes) Find the Common Products 3.OA.7 Count the Square Units 3.MD.6 (7 minutes) (6 minutes) Find the Common Products (7 minutes) Materials: (S) Blank paper Note: This fluency reviews multiplication patterns from G3–Module 3. T: T: T: T: S: T: Fold your paper in half vertically. On the left half, count by threes to 30 down the side of your paper. On the right half, count by sixes to 60 down the side of your paper. Draw a line to match the products that appear in both columns. (Match 6, 12, 18, 24, and 30.) (Write × 3 = 6, × 3 = 12, × 3 = 18, × 3 = 24, and × 3 = 30 next to each matched product on the left half of the paper.) Write the equations next to their products like I did, completing the unknown factors. (Write equations and complete unknowns.) (Write 6 = × 6, 12 = × 6, 18 = × 6, 24 = × 6, and 30 = × 6 next to each matched product on the left half of the paper.) Write the equations next to their products like I did, completing the unknown factors. S: T: Lesson 3: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Model tiling with centimeter and inch unit squares as a strategy to measure area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.24 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 3 S: T: S: T: S: T: S: (Write equations and complete unknowns.) (Write 2 × 3 = × 6.) Say the equation, completing the unknown factor. 2 × 3 = 1 × 6. (Write 2 × 3 = 1 × 6.) Write the remaining equal facts as equations. (Write 4 × 3 = 2 × 6, 6 × 3 = 3 × 6, 8 × 3 = 4 × 6, and 10 × 3 = 5 × 6.) What is the pattern in your equations? Each multiple of 6 is also a multiple of 3. Figures for Count the Square Units Figure 1 Figure 2 Figure 3 Count the Square Units (6 minutes) Note: This fluency reviews finding total area using square units. T: S: (Project a 1 × 5 tiled array similar to Figure 1 at right.) What’s the area of the rectangle? (Pause.) 5 square units. Figure 4 Continue with Figures 2–5. Figure 5 Application Problem (5 minutes) Jace uses paper squares to create a rectangle. Clary cuts all of Jace’s squares in half to create triangles. She uses all the triangles to make a rectangle. There are 16 triangles in Clary’s rectangle. How many squares were in Jace’s shape? Possible student solutions: Dividing Drawing a picture Skip-counting by twos Lesson 3: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Model tiling with centimeter and inch unit squares as a strategy to measure area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.25 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 3 Note: This problem reviews multiplying or dividing by units of 2 from G3 –Module 1, depending on how students solve. Invite students to share their strategies for solving. Concept Development (32 minutes) Materials: (S) Square-centimeter and square-inch tiles (from G3–M4–Lesson 2), centimeter and inch grid paper, ruler, personal white board Pass out 10 square-centimeter tiles to each student. T: Arrange all of your square tiles in 2 equal rows to create a rectangle. Make sure the tiles are touching and don’t overlap. (Allow students time to create rectangle.) What is the area of your rectangle? 10 square units. Is there another way you could arrange all of your tiles to make a rectangle? We could make 5 rows of 2. Or, 1 row of 10. Make 1 row of 10. (Allow students time to make new rectangle.) What is the area of your rectangle now? It’s still 10 square units! NOTES ON Use your ruler to measure all four sides of a tile in MULTIPLE MEANS centimeters. (Wait for students to measure.) Can we OF ACTION AND define these units more precisely? EXPRESSION: Yes, they’re square centimeters! Yes, all four sides Offer an alternative to drawing, measure 1 centimeter so they’re square centimeters. shading, and erasing rectangles using a What is the area of your rectangle in square marker. Some students may find it centimeters? easier to represent and shade rectangles using a Smart Board or 10 square centimeters. personal computer. (Pass out centimeter grid paper.) Slip the grid paper into your personal board. Each side of the square in the grid measures 1 centimeter. How is this grid paper NOTES ON like the tiles we used? MULTIPLE MEANS They’re both square centimeters. OF ACTION Shade the grid paper to represent the rectangle you AND EXPRESSION: made with tiles. Remove a tile from your rectangle, making sure your tiles all still touch to form a rectangle. (Pause.) What is the area of the rectangle now? 9 square centimeters! How can you change the rectangle on the grid paper to have the same area as your new tile rectangle? Erase one of the squares. Support English language learners as they compose their written response to Problem 3. Discussing their reasoning with a partner before writing may be advantageous. Encourage students to use area and square units in their response. Request that the student clarify, if necessary, and guide the elaboration of their ideas. S: T: S: T: S: T: S: T: S: T: S: T: T: S: T: S: Lesson 3: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Model tiling with centimeter and inch unit squares as a strategy to measure area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.26 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 3 T: S: Go ahead and do that. What is the area of the shaded rectangle? 9 square centimeters. Repeat this process with the inch tiles and grid paper. If time allows, students can shade a shape for a partner, who then finds the area of the shape. Then they can erase squares to create shapes with smaller areas. As students are ready, they can start to draw shapes using squares rather than just erasing them. Problem Set (10 minutes) Square-inch and square-centimeter grid paper are needed for some of these problems. Students should do their personal best to complete the Problem Set within the allotted 10 minutes. For some classes, it may be appropriate to modify the assignment by specifying which problems they work on first. Some problems do not specify a method for solving. Students solve these problems using the RDW approach used for Application Problems. Student Debrief (10 minutes) Lesson Objective: Model tiling with centimeter and inch unit squares as a strategy to measure area. The Student Debrief is intended to invite reflection and active processing of the total lesson experience. Invite students to review their solutions for the Problem Set. They should check work by comparing answers with a partner before going over answers as a class. Look for misconceptions or misunderstandings that can be addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the lesson. You may choose to use any combination of the questions below to lead the discussion. How are the rectangles in Problems 1(b) and 1(c) the same? How are they different? How are the rectangles in Problems 1(a) and 2(a) the same? How are they different? Which rectangle in Problem 2 has the biggest area? How do you know? Compare the rectangles you made in Problem 4 with a partner’s rectangles. How are they the same? How are they different? Lesson 3: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Model tiling with centimeter and inch unit squares as a strategy to measure area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.27 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 3 Exit Ticket (3 minutes) After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help you assess the studentsâ€™ understanding of the concepts that were presented in the lesson today and plan more effectively for future lessons. You may read the questions aloud to the students. Examples of Problems 3(b) and 4 Lesson 3: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Model tiling with centimeter and inch unit squares as a strategy to measure area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.28 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 Problem Set 3 Name Date 1. Each is 1 square unit. What is the area of each of the following rectangles? A: square units A B B: _____________________________ C: _____________________________ C D D: _____________________________ 2. Each is 1 square unit. What is the area of each of the following rectangles? a. b. ____________________________ _ c. ____________________________ _ d. ____________________________ _ ____________________________ _ Lesson 3: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Model tiling with centimeter and inch unit squares as a strategy to measure area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.29 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 Problem Set 3 3. a. How would the rectangles in Problem 1 be different if they were composed of square inches? b. Select one rectangle from Problem 1 and recreate it on square-inch and square-centimeter grid paper. 4. Use a separate piece of square-centimeter grid paper. Draw four different rectangles that each has an area of 8 square centimeters. Lesson 3: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Model tiling with centimeter and inch unit squares as a strategy to measure area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.30 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 Exit Ticket 3 Name Date 1. Each is 1 square unit. Write the area of Rectangle A. Then draw another rectangle with the same area in the space provided. A Area = ____________________________ 2. Each is 1 square unit. Does this rectangle have the same area as Rectangle A? Explain. Lesson 3: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Model tiling with centimeter and inch unit squares as a strategy to measure area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.31 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 Homework 3â€˘ 4 Name 1. Each Date is 1 square unit. What is the area of each of the following rectangles? A: A B square units B: __________________ C: __________________ C D D: __________________ 2. Each is 1 square unit. What is the area of each of the following rectangles? a. b. c. d. Lesson 3: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Model tiling with centimeter and inch unit squares as a strategy to measure area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.32 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 Homework 3â€˘ 4 3. Each is 1 square unit. Write the area of each rectangle. Then draw another rectangle with the same area in the space provided. A Area = square units B Area = __________________________ C Area = __________________________ Lesson 3: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Model tiling with centimeter and inch unit squares as a strategy to measure area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.33 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 Template 3â€˘ 4 Lesson 3: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Model tiling with centimeter and inch unit squares as a strategy to measure area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.34 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 Template 3â€˘ 4 Lesson 3: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Model tiling with centimeter and inch unit squares as a strategy to measure area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.35 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 3• 4 Lesson 4 Objective: Relate side lengths with the number of tiles on a side. Suggested Lesson Structure Fluency Practice Application Problem Concept Development Student Debrief Total Time (12 minutes) (5 minutes) (33 minutes) (10 minutes) (60 minutes) Fluency Practice (12 minutes) Group Counting 3.OA.1 Products in an Array 3.OA.3 Count the Square Units 3.MD.6 (3 minutes) (3 minutes) (6 minutes) Group Counting (3 minutes) Note: Group counting reviews interpreting multiplication as repeated addition. Direct students to count forward and backward, occasionally changing the direction of the count. Sixes to 60 Sevens to 70 Eights to 80 Nines to 90 Products in an Array (3 minutes) Materials: (S) Personal white boards Note: This fluency anticipates relating multiplication with area in G3–M4–Topic B. T: S: T: S: T: (Project an array with 5 rows of 3 stars.) How many rows of stars do you see? 5 rows. How many stars are in each row? 3 stars. On your boards, write two multiplication sentences that can be used to find the total number of stars. Lesson 4: Date: Relate side lengths with the number of tiles on a side. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.36 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 3• 4 S: (Write 5 × 3 = 15 and 3 × 5 = 15.) Continue with the following possible sequence: 4 by 6, 7 by 3, 8 by 5, and 9 by 7. Count the Square Units (6 minutes) Note: This fluency reviews comparing the area of different shapes. T: S: T: S: T: S: T: S: T: S: T: S: (Project an 8 × 1 tiled array.) How many square units are in the rectangle? 8 units. (Write 8 units next to the rectangle. Project a 4 × 2 tiled array.) How many square units are in the rectangle? 8 units. (Write 8 units next to the rectangle. Project a 2 × 4 tiled array.) How many square units are in the rectangle? 8 units. (Write 8 units next to the rectangle. Project a 1 × 8 tiled array.) How many square units are in the rectangle? 8 units. (Write 8 units next to the rectangle.) Do the four rectangles look the same? No. What do the rectangles have in common? They are each made up of 8 square units. Continue with the following possible sequence: 12 × 1, 1 × 12, 6 × 2, 3 × 4, 2 × 6, and 4 × 3. Application Problem (5 minutes) Mara uses 15 square-centimeter tiles to make a rectangle. Ashton uses 9 square-centimeter tiles to make a rectangle. a. Draw what Mara’s and Ashton’s rectangles might look like. b. Whose rectangle has a bigger area? How do you know? Note: This problem reviews G3–M4–Lesson 2, specifically tiling with square units. Invite students to share and compare their drawings for Mara’s and Ashton’s rectangles. Lesson 4: Date: Relate side lengths with the number of tiles on a side. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.37 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 3• 4 Concept Development (33 minutes) Materials: (S) 15 square-inch and square-centimeter tiles, ruler, personal white board Pass out 15 square-inch tiles to each student. T: S: T: S: T: These tiles are square…? Inches! Use the tiles to make a 3 by 5 array. (Allow students time to make array.) Push the tiles together to form a rectangle with no gaps or overlaps. What is the area of your rectangle? 15 square inches. I see your squares are nicely arranged to form a rectangle. What about these? (Project Rectangles A and B shown at Rectangle A right.) I used 15 square-inch tiles to make both of these rectangles. Talk to a partner. Is the area of both rectangles 15 square inches? Yes, the number of tiles is the same. No, A’s area is bigger than 15 square inches because there are gaps between the tiles. B’s area is smaller because some of the tiles are on top of each other. Why is it important to avoid gaps or overlaps when we measure area? If there are gaps or overlaps the amount of space the rectangle takes up changes. The square unit would be wrong since Rectangle B some area is taken away if there are overlaps or some is added if there are gaps. Use your ruler to measure across the top of your rectangle in inches. What is the length of this side? 5 inches. How many tiles are on this side? 5 tiles. Use your ruler to measure the shorter side of the rectangle in inches. What is the length of this side? 3 inches. How many tiles are on this side? NOTES ON 3 tiles! MULTIPLE MEANS What is the relationship between the number of tiles OF REPRESENTATION: on a side and the side length of the rectangle? Scaffold student contrast of length and They’re the same! area. Consider placing a long string along the side of the rectangle, or have What do you notice about the lengths of the opposite students trace the side with a finger to sides of the rectangles? better illustrate length. In contrast, They are equal! have students shade in the area before writing 15 square inches. S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: MP.8 Lesson 4: Date: Relate side lengths with the number of tiles on a side. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.38 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 3• 4 T: S: T: S: T: S: T: Trace the rectangle on your board, then remove the tiles and label the side lengths. Now write the area inside the rectangle. What are the units for the side lengths? Inches. What are the units for the area? Square inches. Talk to a partner, why are the units different for side lengths and area? The unit for side lengths is inches because we used a ruler to measure the length of the side in inches. For area, the unit is square inches because we counted the number of square-inch tiles that we used to make the rectangle. Inches are used to measure lengths, like the side lengths, and square inches are used to measure the amount of flat space a figure takes up, which is the area. These tiles are square… Centimeters! Use them to make a rectangle with side lengths of 5 centimeters and 4 centimeters. (Write 5 cm and 4 cm.) Tell your partner how many tiles you’ll count to make each side. I’ll make one side with 5 tiles and the other with 4 tiles. Actually we’ll count 5 tiles each for two sides of the rectangle, and 4 tiles each for the other two sides. Opposite sides are the same, remember? Make your rectangle on top of your personal board. Label the side lengths. (Make rectangle and label side lengths 5 cm and 4 cm.) NOTES ON MULTIPLE MEANS OF How many fives did you make? Why? REPRESENTATION: 4 fives, because the other side length is 4. Alternatively, build the rectangle in 4 What is the total of 4 fives? rows of 5 centimeter tiles. As students 20. place each row, encourage careful and meaningful counting. Students may Skip-count your fives to find the total area of the benefit from counting each tile in the rectangle. (Pause.) What is the total area? row so as not to add extra tiles. Then, 20 square centimeters! recapture by counting by fives, “5, 10, What is the relationship between the side lengths and 15, 20.” area? If you multiply 5 times 4 then you get 20! Direct students to exchange square-inch tiles for square-centimeter tiles. T: S: T: S: T: S: T: S: T: S: T: S: T: S: If time allows, repeat the process with a rectangle with side lengths of 3 centimeters and 6 centimeters. As students are ready, tell them the area and let them build a rectangle and name the side lengths. Lesson 4: Date: Relate side lengths with the number of tiles on a side. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.39 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 3• 4 Problem Set (10 minutes) Students should do their personal best to complete the Problem Set within the allotted 10 minutes. For some classes, it may be appropriate to modify the assignment by specifying which problems they work on first. Some problems do not specify a method for solving. Students solve these problems using the RDW approach used for Application Problems. Student Debrief (10 minutes) Lesson Objective: Relate side lengths with the number of tiles on a side. The Student Debrief is intended to invite reflection and active processing of the total lesson experience. Invite students to review their solutions for the Problem Set. They should check work by comparing answers with a partner before going over answers as a class. Look for misconceptions or misunderstandings that can be addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the lesson. You may choose to use any combination of the questions below to lead the discussion. Tell a partner how you could use squarecentimeter tiles to check your work in Problem 1. Compare the areas of the rectangles in Problems 1 and 2. Which rectangle has a bigger area? How do you know? What are the side lengths of the shape in Problem 3? Are all the sides the same? How do you know? What shape is this? What is the area of the rectangle in Problem 4? Explain how you found the area to a partner. How many centimeter tiles fit in the rectangle in Problem 5? Is that the area of the rectangle in square centimeters? Why or why not? In Problem 6, if the side length of A is 4 units, would 3 units make sense for the side length of B? Why or why not? What would make sense? Lesson 4: Date: Relate side lengths with the number of tiles on a side. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.40 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 3• 4 Exit Ticket (3 minutes) After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help you assess the students’ understanding of the concepts that were presented in the lesson today and plan more effectively for future lessons. You may read the questions aloud to the student. Lesson 4: Date: Relate side lengths with the number of tiles on a side. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.41 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 Problem Set 3â€˘ 4 Name Date 1. Use a ruler to measure the side lengths of the rectangle in centimeters. Mark each centimeter with a point and connect the points to show the square units. Then count the squares you drew to find the total area. Total area: _____________________________________ 2. Use a ruler to measure the side lengths of the rectangle in inches. Mark each inch with a point and connect the points to show the square units. Then count the squares you drew to find the total area. Total area: _____________________________________ 3. Mariana uses square-centimeter tiles to find the side lengths of the rectangle below. Label each side length. Then count the tiles to find the total area. Total area: _____________________________________ Lesson 4: Date: Relate side lengths with the number of tiles on a side. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.42 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 Problem Set 3â€˘ 4 4. Each is 1 square centimeter. Saffron says that the side length of the rectangle below is 4 centimeters. Kevin says the side length is 5 centimeters. Who is correct? Explain how you know. 5. Use both square-centimeter and square-inch tiles to find the area of the rectangle below. Which works best? Explain why. 6. How does knowing side lengths A and B help you find side lengths C and D on the rectangle below? B A C D Lesson 4: Date: Relate side lengths with the number of tiles on a side. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.43 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 Exit Ticket 3• 4 Name Date Label the side lengths of each rectangle. Then match the rectangle to its total area. 12 sq cm 5 sq in 6 sq cm Lesson 4: Date: Relate side lengths with the number of tiles on a side. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.44 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 Homework 3â€˘ 4 Name Date 1. Ella placed square-centimeter tiles on the rectangle below, and then labeled the side lengths. What is the area of her rectangle? 4 cm 2 cm Total area: ________________________ 2. Kyle uses square-centimeter tiles to find the side lengths of the rectangle below. Label each side length. Then count the tiles to find the total area. Total area: ________________________ 3. Maura uses square-inch tiles to find the side lengths of the rectangle below. Label each side length. Then find the total area. Total area: ________________________ Lesson 4: Date: Relate side lengths with the number of tiles on a side. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.45 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 Homework 3â€˘ 4 4. Each square unit below is 1 square inch. Claire says that the side length of the rectangle below is 3 inches. Tyler says the side length is 5 inches. Who is correct? Explain how you know. 5. Label the unknown side lengths for the rectangle below, then find the area. Explain how you used the lengths provided to find the unknown lengths and area. 4 inches 2 inches _________________ _________________ Lesson 4: Date: Relate side lengths with the number of tiles on a side. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.A.46 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org New York State Common Core GRADE 3 Mathematics Curriculum GRADE 3 • MODULE 4 Topic B Concepts of Area Measurement 3.MD.5, 3.MD.6, 3.MD.7 Focus Standards: 3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). Relate area to the operations of multiplication and addition. a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent wholenumber products as rectangular areas in mathematical reasoning. d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems. Addition and Subtraction of Length Units Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 Multiplication and Division with Units of 0, 1, 6–9, and Multiples of 10 Multi-Digit Multiplication and Division Exploring Multiplication 3.MD.6 3.MD.7 Instructional Days: Coherence -Links from: 4 G2–M2 G3–M1 G3–M3 -Links to: G4–M3 G4–M7 In previous lessons, students tiled given rectangles. In Lesson 5, students build rectangles using unit square tiles to make arrays when given specific criteria. For example, students may be told that there are 24 tiles inside the rectangle and that one side of the rectangle is covered with 4 tiles. Students may start by building one column of the array to represent a length of 4 units, then duplicate that process until they reach 24 total tiles, skip-counting by fours. Finally they physically push together the rows of tiles to make the array. When they count the number of fours, the process connects to unknown factor problems (in this case, the unknown factor of 6) from previous modules and builds toward students’ discovery of the area formula. Now experienced with drawing rectangular arrays within an area model, students find the area of an Topic B: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Concepts of Area Measurement 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported.License. 4.B.1 NYS COMMON CORE MATHEMATICS CURRICULUM Topic B 3â€˘ 4 incomplete array in Lesson 6. They visualize and predict what the finished array looks like, then complete it by joining opposite end points with a straight edge and determine the total area using skip-counting. The incomplete array model bridges to the area model, where no array is given. In Lesson 7, students are given information about the side lengths of an area model (shown at right). Based on this information they use a straight edge to draw a grid of equal sized squares within the area model, then skip-count to find the total number of squares. Units move beyond square centimeters and inches to include square feet and square meters. Array Area Model In Lesson 8, students recognize that side lengths play an important part in determining the area of a rectangle. They understand that multiplying the number of square units in a row by the number of rows produces the same result as skip-counting the squares within the array. Given the area and one side length, students realize that they can use multiplication with an unknown factor or division to find the unknown side length. A Teaching Sequence Towards Mastery of Concepts of Area Measurement Objective 1: Form rectangles by tiling with unit squares to make arrays. (Lesson 5) Objective 2: Draw rows and columns to determine the area of a rectangle, given an incomplete array. (Lesson 6) Objective 3: Interpret area models to form rectangular arrays. (Lesson 7) Objective 4: Find the area of a rectangle through multiplication of the side lengths. (Lesson 8) Topic B: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Concepts of Area Measurement 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported.License. 4.B.2 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 3 Lesson 5 Objective: Form rectangles by tiling with unit squares to make arrays. Suggested Lesson Structure Fluency Practice Application Problem Concept Development Student Debrief Total Time (14 minutes) (6 minutes) (30 minutes) (10 minutes) (60 minutes) Fluency Practice (14 minutes) Group Counting 3.OA.1 Products in an Array 3.OA.3 Find the Common Products 3.OA.7 (3 minutes) (3 minutes) (8 minutes) Group Counting (3 minutes) Note: Group counting reviews interpreting multiplication as repeated addition. Direct students to count forward and backward, occasionally changing the direction of the count. Threes to 30 Sixes to 60 Sevens to 70 Nines to 90 Products in an Array (3 minutes) Materials: (S) Personal white boards Note: This fluency anticipates relating multiplication with area in G3–M4–Topic B. T: S: T: S: T: (Project an array with 4 rows of 3 stars.) How many rows of stars do you see? 4 rows. How many stars are in each row? 3 stars. On your boards, write two multiplication sentences that can be used to find the total number of stars. Lesson 5: Date: Form rectangles by tiling with unit squares to make arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.3 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 3 S: (Write 4 × 3 = 12 and 3 × 4 = 12.) Continue with the following possible sequence: 3 × 6, 7 × 5, 8 × 4, and 9 × 6. Find the Common Products (8 minutes) Materials: (S) Blank paper Note: This fluency reviews multiplication patterns from G3–Module 3. T: S: T: (List the multiples of 4 and 8.) Draw a line to match the numbers that appear in both columns. (Match 8, 16, 24, 32, and 40.) (Write 2 × 4 = 8, etc., next to each matched number on the left half of the paper.) Write the rest of the number sentences like I did. (Write 8 = 1 × 8, etc., next to each matched number on the right half of the paper.) Write the rest of the equations like I did. (Write equations.) (Write 4 × 2 = × 8.) Say the true equation. 2 × 4 = 1 × 8. (Write 2 × 4 = 1 × 8.) Write the remaining equal facts as equations. (Write 4 × 4 = 2 × 8, 6 × 4 = 3 × 8, 8 × 4 = 4 × 8, 10 × 4 = 5 × 8.) Discuss the patterns in your equations. Each multiple of 8 is also a multiple of 4. T: S: T: S: T: S: T: S: Application Problem (6 minutes) Candice uses square-centimeter tiles to find the side lengths of a rectangle as shown. She says the side lengths are 5 centimeters and 7 centimeters. Her partner, Luis uses a ruler to check Candice’s work and says that the side lengths are 5 centimeters and 6 centimeters. Who is right? How do you know? Lesson 5: Date: Form rectangles by tiling with unit squares to make arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.4 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 3 Note: This problem reviews G3–M4–Lesson 4, specifically the relationship between the number of tiles and the side length. Invite the students to discuss what Luis might have done wrong. Concept Development (30 minutes) Materials: (S) 15 square-inch tiles per student, personal white board, straight edge, blank paper Concrete: Understand the relationship between side lengths and area. (Draw or project the rectangle and side length shown to the right.) T: S: T: S: T: S: T: S: T: S: T: 2 in Use square-inch tiles to show this rectangle as an array. What information do we know? There are 2 rows. A side length is 2 inches. At your table, place tiles to make the known side. NOTES ON (Make 1 column of 2 tiles.) MULTIPLE MEANS OF (Write below the diagram: Area = 12 sq in.) How many REPRESENTATION: total tiles will we use to make our rectangle? Simplify and clarify your script for 12 tiles. English language learners and others. Rephrase, “What information do we How many twos are in 12? know?” to “How many rows of inch 6 twos. squares? How do you know?” Use your tiles to make 6 twos, then skip-count to check Place the opaque 2 by 6 rectangle your work. pictured above over a square grid. As (Make 6 groups of 2 tiles and skip-count.) 2, 4, 6, 8, 10, an alternate way to check work, lift the 12. rectangle to show the 12 squares covered. Push your twos together to make a rectangle. (After students do so, add a question mark to the diagram as shown at right.) What is the unknown side length? Six. Six tiles. Six inches. ? (Replace the question mark with 6 in on the diagram.) Tell your 2 in partner about the relationship between the side lengths and the area. Write an equation to show your thinking. Be sure to include the units. Area = 12 sq in 2 inches × 6 inches = 12 square inches, so the area is the product of the side lengths. (Write 2 in × 6 in = 12 sq in.) S: T: S: Repeat the process using a rectangle with a known side length of 5 inches and an area of 15 square inches. Ask students to write an unknown factor problem, 5 × = 15, then use the tiles to solve. Lesson 5: Date: Form rectangles by tiling with unit squares to make arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.5 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 3 Concrete/Pictorial: Form rectangles and determine area or side lengths by drawing to make arrays. T: Lay tiles on your personal board to make a side 3 inches tall. Trace the outline of all 3 tiles. Then, draw horizontal 7 in lines to show where they connect. (Draw image shown at right.) 3 in Label the side length. (Label 3 in, as shown.) Use your tiles to make another side, 7 inches long. (Add tiles horizontally, using the corner tile as one of the 7.) Trace the outline of the tiles. Draw vertical lines to show where they connect. Label the side length. (Drawing shown to the right, label 7 in as shown.) How many threes will be in this rectangle? 7 threes. Talk to your partner. Which strategy might you use to find the total area of the rectangle? We can draw in the rest of the squares and count them all. Or, just skip-count 7 threes. It would be easier to just multiply 7 inches × 3 inches and get 21 square inches. Many students suggested multiplying the side lengths to find the area. Let’s check this strategy by drawing in the rest of squares. Use your straight edge to draw the rest of the tiles in the rectangle, then skip-count to find the total area. (Follow the grid lines to make the other tiles, then skip-count.) 3, 6, 9, 12, 15, 18, 21. Does 7 inches × 3 inches = 21 square inches accurately give the area of the rectangle? Yes! 6 in Clear your board and use your tiles to make a side length of 6 inches. Trace the outline of all 6 tiles. Then draw horizontal lines to show where they connect. (Draw image shown at right.) Label the side length. (Label 6 in, as shown.) Write 6 × __ = 24 on your board. Talk to a partner, how can you use this equation to help you find the other side length? From the equation, I know that the area is 24, so I can add rows of 6 tiles until I have 24 tiles. Then, I can count the rows to find the side length. I can skip-count by 6 to get to 24, and then I know the other side length will be equal to the number of times I skip-count. I know 6 × 4 = 24, so I know that the other side length is 4. Choose a strategy to find the other side length and then fill in the blank in the equation. (Allow time for students to work.) What is the other side length? 4 inches! S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: Lesson 5: Date: Form rectangles by tiling with unit squares to make arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.6 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 3 Problem Set (10 minutes) Students should do their personal best to complete the Problem Set within the allotted 10 minutes. For some classes, it may be appropriate to modify the assignment by specifying which problems they work on first. Some problems do not specify a method for solving. Students solve these problems using the RDW approach used for Application Problems. NOTES ON MULTIPLE MEANS OF ACTION AND EXPRESSION: Some learners may benefit from alternatives to drawing tiles inside rectangles on the Problem Set. Consider the following: Magnify the worksheet to ease small motor tasks. Provide virtual or concrete manipulatives. Allow students to draw their own rectangles, perhaps with larger tiles, perhaps with smaller areas. Student Debrief (10 minutes) Lesson Objective: Form rectangles by tiling with unit squares to make arrays. The Student Debrief is intended to invite reflection and active processing of the total lesson experience. Invite students to review their solutions for the Problem Set. They should check work by comparing answers with a partner before going over answers as a class. Look for misconceptions or misunderstandings that can be addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the lesson. You may choose to use any combination of the questions below to lead the discussion. Compare Problems 1(b) and 1(e), and Problems 1(a) and 1(c). How does each pair show commutativity? How many more threes does the array in Problem 1(d) have than the array in Problem 1(a)? How might the side lengths help you know that, even without seeing the tiled array? Compare Problems 1(c) and 1(f). How are the areas related? (The area of 1(f) is half the area of 1(c).) How might you have figured that out just by knowing the side lengths of each array? Students may have different solutions for Problem 3. Invite them to share and compare their work. In Problem 2, what strategy did you use to find the unknown side length? Is there another way you could have figured it out? MP.8 Lesson 5: Date: Form rectangles by tiling with unit squares to make arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.7 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 3 Exit Ticket (3 minutes) After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help you assess the studentsâ€™ understanding of the concepts that were presented in the lesson today and plan more effectively for future lessons. You may read the questions aloud to the students. Lesson 5: Date: Form rectangles by tiling with unit squares to make arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.8 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 Problem Set 3 Name Date 1. Use the centimeter side of a ruler to draw in the tiles, then skip-count to find the unknown side length or area. Write a multiplication sentence for each tiled rectangle. a. Area: 18 square centimeters. d. Area: 24 square centimeters. 3 cm 3 cm 3 6 9 12 15 18 3 18 _______ × _______ = _______ b. Area: _____ square centimeters. 5 cm _______ × _______ = _______ e. Area: 20 square centimeters. 4 cm 5 cm _______ × _______ = _______ c. Area: 18 square centimeters. f. _______ × _______ = _______ Area: _____ square centimeters. 3 cm 6 cm _______ × _______ = _______ 3 cm _______ × _______ = _______ Lesson 5: Date: Form rectangles by tiling with unit squares to make arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.9 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 Problem Set 3â€˘ 4 2. Lindsey makes a rectangle with 35 square-inch tiles. She arranges the tiles in 5 equal rows. What are the side lengths of the rectangle? Use words, pictures, and numbers to support your answer. 3. Mark has a total of 24 square-inch tiles. He uses 18 square-inch tiles to build one rectangular array. He uses the remaining square-inch tiles to build a second rectangular array. Draw two arrays that Mark might have made. Then write multiplication sentences for each. 4. Leon makes a rectangle with 32 square-centimeter tiles. There are 4 equal rows of tiles. a. How many tiles are in each row? Use words, pictures, and numbers to support your answer. b. Can Leon arrange all of his 32 square-centimeter tiles into 6 equal rows? Explain your answer. Lesson 5: Date: Form rectangles by tiling with unit squares to make arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.10 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 Exit Ticket 3• 4 Name Date Darren has a total of 28 square-centimeter tiles. He arranges them into 7 equal rows. Draw Darren’s rectangle. Label the side lengths, and write a multiplication equation to find the total area. Lesson 5: Date: Form rectangles by tiling with unit squares to make arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.11 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 Homework 3• 4 Name Date 1. Use the centimeter side of a ruler to draw in the tiles, then skip-count to find the unknown side length or area. Write a multiplication sentence for each tiled rectangle. a. Area: 24 square centimeters. b. Area: 24 square centimeters. 4 cm 6 cm 4 24 _______ × _______ = _______ c. Area: 15 square centimeters. _______ × _______ = _______ d. Area: 15 square centimeters. 5 cm 3 cm _______ × _______ = _______ _______ × _______ = _______ Lesson 5: Date: Form rectangles by tiling with unit squares to make arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.12 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 Homework 3â€˘ 4 2. Ally makes a rectangle with 45 square-inch tiles. She arranges the tiles in 5 equal rows. How many square-inch tiles are in each row? Use words, pictures, and numbers to support your answer. 3. Leon makes a rectangle with 36 square-centimeter tiles. There are 4 equal rows of tiles. a. How many tiles are in each row? Use words, pictures, and numbers to support your answer. b. Can Leon arrange all of his 36 square-centimeter tiles into 6 equal rows? Use words, pictures, and numbers to support your answer. c. Do the rectangles in (a) and (b) have the same total area? Explain how you know. Lesson 5: Date: Form rectangles by tiling with unit squares to make arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.13 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 3 Lesson 6 Objective: Draw rows and columns to determine the area of a rectangle, given an incomplete array. Suggested Lesson Structure Fluency Practice Application Problem Concept Development Student Debrief Total Time (12 minutes) (8 minutes) (30 minutes) (10 minutes) (60 minutes) Fluency Practice (12 minutes) Group Counting 3.OA.1 Write the Multiplication Fact 3.MD.7 Products in an Array 3.OA.3 (4 minutes) (4 minutes) (4 minutes) Group Counting (4 minutes) Note: Group counting reviews interpreting multiplication as repeated addition. Direct students to count forward and backward, occasionally changing the direction of the count. Sixes to 60 Sevens to 70 Eights to 80 Nines to 90 Write the Multiplication Fact (4 minutes) Materials: (S) Personal white boards Note: This fluency reviews relating multiplication with area from G3–M4–Lesson 5. T: S: T: S: (Project a 5 by 3 square-unit tiled rectangle. Write × = 15.) There are 15 tiles altogether. How many rows are there? 5 rows. (Write 5 × = 15.) On your boards, fill in the blank to make a true equation. (Write 5 × 3 = 15.) Lesson 6: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Draw rows and columns to determine the area of a rectangle, given an incomplete array. 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.14 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 3 T: S: T: S: (Project a 3 by 4 square-unit tiled rectangle. Write × = 12.) There are 12 tiles altogether. How many columns are there? 4 columns. (Write × 4 = 12.) On your boards, fill in the blank to make a true equation. (Write 3 × 4 = 12.) Continue with the following possible sequence, asking the students to first name either the number of rows or the number of columns: 4 by 6, 6 by 7, 5 by 8, and 7 by 8. Products in an Array (4 minutes) Materials: (S) Personal white boards Note: This fluency supports the relationship between multiplication and area. T: S: T: S: T: S: (Project an array with 2 rows of 6 stars.) How many rows of stars do you see? 2 rows. How many stars are in each row? 6 stars. On your boards, write two multiplication sentences that can be used to find the total number of stars. (Write 2 × 6 = 12 and 6 × 2 = 12.) Continue with the following possible sequence: 3 by 7, 6 by 5, 8 by 6, and 4 by 9. Application Problem (8 minutes) Huma has 4 bags of square-inch tiles with 6 tiles in each bag. She uses them to measure the area of a rectangle on her homework. After covering the rectangle, Huma has 4 tiles left. What is the area of the rectangle? NOTES ON MULTIPLE MEANS OF ENGAGEMENT: Adjust the numbers in the Application Problem to challenge students working above grade level. Note: This problem reviews multi-step word problems in the context of using square tiles to measure area. Lesson 6: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Draw rows and columns to determine the area of a rectangle, given an incomplete array. 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.15 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 3 Concept Development (30 minutes) Materials: (S) Personal white board, straight edge, Problem Set, array template Part 1: Estimate to draw the missing square units inside an array. Students have the array template in their personal boards, looking at Array 1. T: S: How can an array of square units help you find the area of a rectangle? You can count the total number of square units inside the rectangle. You can skip-count the rows to find the total. (Project or display the image at right.) What do you notice about the array inside of this rectangle? Some of the square units are missing. What do you notice about the top row? It has 4 square units and a rectangle. Look at the second row. Can you use those square units to help you know how many square units make the top row? The second row has 1 more square unit than the top row. You can just follow the line it makes to divide the rectangle into 2 square units. Use your straight edge to draw that line now. (Draw as shown at right.) Talk to your partner: Use the top row to figure out how many square units will fit in each of the rows below. How do you know? Each row should have 6 square units, because rows in an array are equal! Use the lines that are already there as guides, and with your straight edge, draw lines to complete the array. (Draw.) How many rows of 6 are in this array? 4 rows of 6. What equation can be used to find the area of the rectangle? 4 × 6 = 24. NOTES ON MULTIPLE MEANS FOR ACTION AND EXPRESSION: Scaffold the following sequence further by beginning with a basic 2 by 2 rectangle in which 2 tiles are missing. Graduate to a 2 by 3 rectangle in which tiles or lines are missing. Continue step by step until students are ready for rectangles with larger areas. Also consider adding color to alternating tiles to assist counting or to distinguish tiles from rectangles or blank space. T: S: T: S: T: Array 1 S: Array 1: Top row complete T: S: T: MP.2 S: T: S: T: S: T: S: Array 1: Fully drawn Lesson 6: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Draw rows and columns to determine the area of a rectangle, given an incomplete array. 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.16 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 3 Part 2: Draw rows and columns to determine the area. T: (Project the rectangle shown at right.) Turn your array template over. Can we estimate to draw unit squares inside the rectangle? Yes. It might take us longer, because fewer units are given. A quicker way to find the area is to figure out the number of rows and the number of columns. Let’s start by finding the number of rows in our array. How can we find the number of rows? The first column shows you how many rows there are. With your finger, show your partner what you’ll draw to find the number of rows. Then draw. (Show and draw.) How can we find the number of columns? The first row shows you how many columns there are. Use your straight edge to complete the first row. Label the side lengths of the rectangle, including units. (Draw and label side lengths 5 units and 6 units.) What number sentence can be used to find the area? 5 × 6 = 30. Array 2 S: T: S: T: S: T: S: T: S: T: S: Array 2: 1 row and 1 column drawn Problem Set (10 minutes) Students should do their personal best to complete the Problem Set within the allotted 10 minutes. For some classes, it may be appropriate to modify the assignment by specifying which problems they work on first. Some problems do not specify a method for solving. Students solve these problems using the RDW approach used for Application Problems. Student Debrief (10 minutes) Lesson Objective: Draw rows and columns to determine the area of a rectangle, given an incomplete array. The Student Debrief is intended to invite reflection and active processing of the total lesson experience. Invite students to review their solutions for the Problem Set. They should check work by comparing answers with a partner before going over answers as a class. Look for Lesson 6: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Draw rows and columns to determine the area of a rectangle, given an incomplete array. 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.17 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 3 misconceptions or misunderstandings that can be addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the lesson. You may choose to use any combination of the questions below to lead the discussion. How did you know where to draw the columns and rows in Problem 1? To find area, why don’t we need to draw all of the square units in an incomplete array? What mistake did Sheena make in Problem 2? Is it necessary to have the rug to solve Problem 3? Why or why not? In Problem 3, how many tiles does the rug touch? There are multiple ways to find a solution to Problem 4. Invite students to share how they found the answer. Exit Ticket (3 minutes) After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help you assess the students’ understanding of the concepts that were presented in the lesson today and plan more effectively for future lessons. You may read the questions aloud to the students. Lesson 6: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Draw rows and columns to determine the area of a rectangle, given an incomplete array. 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.18 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 Problem Set 3 Name Date 1. Each represents a 1-cm square. Draw to find the number of rows and columns in each array. Match it to its completed array. Then fill in the blanks to make a true equation to find each array’s area. a. _____ × _____ = _____ sq cm b. _____ × _____ = _____ sq cm c. _____ × _____ = _____ sq cm d. _____ × _____ = _____ sq cm e. _____ × _____ = _____ sq cm f. _____ × _____ = _____ sq cm Lesson 6: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Draw rows and columns to determine the area of a rectangle, given an incomplete array. 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.19 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 Problem Set 3 2. Sheena skip-counts by sixes to find the total square units in the rectangle below. She says there are 42 square units. Is she right? Explain your answer. 3. The tile floor in Brandonâ€™s living room has a rug on it as shown below. How many square tiles are on the floor, including the tiles under the rug? 4. Abdul is creating a stained glass window with square-inch glass tiles as shown below. How many more square-inch glass tiles does Abdul need to finish his glass window? Explain your answer. Lesson 6: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Draw rows and columns to determine the area of a rectangle, given an incomplete array. 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.20 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 Exit Ticket 3 Name Date The tiled floor in Caydenâ€™s dining room has a rug on it as shown below. How many square tiles are on the floor, including the tiles under the rug? Lesson 6: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Draw rows and columns to determine the area of a rectangle, given an incomplete array. 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.21 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 Homework 3 Name Date 1. Each represents a 1-cm square. Draw to find the number of rows and columns in each array. Match it to its completed array. Then fill in the blanks to make a true equation to find each array’s area. a. _____ × _____ = _____ sq cm b. _____ × _____ = _____ sq cm c. _____ × _____ = _____ sq cm d. _____ × _____ = _____ sq cm e. _____ × _____ = _____ sq cm f. _____ × _____ = _____ sq cm Lesson 6: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Draw rows and columns to determine the area of a rectangle, given an incomplete array. 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.22 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 Homework 3 2. Minh skip-counts by sixes to find the total square units in the rectangle below. She says there are 36 square units. Is she correct? Explain your answer. 3. The tub in Paigeâ€™s bathroom covers the tile floor as shown below. How many square tiles are on the floor, including the tiles under the tub? 4. Frank sees a book on top of his chessboard. How many squares are covered by the book? Explain your answer. Lesson 6: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Draw rows and columns to determine the area of a rectangle, given an incomplete array. 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.23 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 Array Template 3 Array 1 Lesson 6: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Draw rows and columns to determine the area of a rectangle, given an incomplete array. 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.24 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 Array Template 3 Array 2 Lesson 6: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Draw rows and columns to determine the area of a rectangle, given an incomplete array. 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.25 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 3• 4 Lesson 7 Objective: Interpret area models to form rectangular arrays. Suggested Lesson Structure Fluency Practice Application Problem Concept Development Student Debrief Total Time (12 minutes) (8 minutes) (30 minutes) (10 minutes) (60 minutes) Fluency Practice (12 minutes) Group Counting 3.OA.1 Draw Rectangles 3.MD.5 Draw Rectangular Arrays 3.MD.5 (4 minutes) (4 minutes) (4 minutes) Group Counting (4 minutes) Note: Group counting reviews interpreting multiplication as repeated addition. Direct students to count forward and backward, occasionally changing the direction of the count. Sixes to 60 Sevens to 70 Eights to 80 Nines to 90 Draw Rectangles (4 minutes) Materials: (S) Grid paper Note: This fluency reviews drawing a rectangle from a known area. Show student work that is correct, but looks different (e.g., a 6 × 2 unit rectangle juxtaposed with a 4 × 3 unit rectangle). T: S: Draw a rectangle that has an area of 6 square units. (Draw a 6 square unit rectangle.) Continue with the following possible sequence: 10 square units, 12 square units, 16 square units, 24 square units, and 35 square units. Lesson 7: Date: Interpret area models to form rectangular arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.26 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 3• 4 Draw Rectangular Arrays (4 minutes) Materials: (S) Grid paper Note: This fluency reviews finding area using side lengths. T: T: S: Draw a 4 × 2 rectangular array using the squares on your grid paper. How many square units are in your array? 8 square units. Continue with the following possible sequence: 6 × 2 units, 4 × 3 units, 6 × 3 units, 9 × 2 units, 6 × 4 units, and 3 × 8 units. Application Problem (8 minutes) Lori wants to replace the square tiles on her wall. The square tiles are sold in boxes of 8 square tiles. Lori buys 6 boxes of tiles. Does she have enough to replace all the tiles, including the tiles under the painting? Explain your answer. Painting Note: This problem reviews multi-step word problems in the context of using square tiles to measure area. It also reviews finding the array of an incomplete array from G3–M4–Lesson 6. Concept Development (30 minutes) Materials: (T) Meter stick, 12-inch ruler, pad of square sticky notes (S) 1 set per pair of 12 square-inch and 12 square-centimeter paper tiles from G3–M4–Lesson 2, personal white boards, rulers, area model template Part 1: Explore the relationship between units and area. T: One partner will use square inches, and the other will use square centimeters. Work together to decide on how to arrange your tiles to make the same shape rectangle. Then make that rectangle with your pieces. Lesson 7: Date: Interpret area models to form rectangular arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.27 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 3• 4 S: T: S: T: S: T: MP.6 S: T: S: T: S: T: S: (Decide on a rectangle and represent it using square inches and square centimeters.) You and your partner each made the same shape rectangle. Is the area also the same? Yes, because we both used the same number of pieces. Yeah, but my pieces are smaller than yours. They’re square centimeters, and look, my shape takes up less space on the table. The area of the shape with square inches is bigger because inches are bigger than centimeters. Turn your personal board horizontal and write the area of your rectangle. (Write either 12 sq in or 12 sq cm.) (Draw 1 square meter on the board.) This is 1 square meter. Suppose you used 12 square-meter tiles to make your rectangle instead. Would this rectangle have a larger area or a smaller area than your original rectangle? It would be much larger! (Draw 1 square foot on the board.) How would your rectangle compare if you made it from 12 square feet? It would be bigger than 12 square inches or centimeters, but smaller than 12 square meters. (Hold up a pad of square sticky notes.) How about if you had used 12 sticky notes? Still bigger than 12 square inches or centimeters, but smaller than 12 square feet or meters. Why is it important to label the unit when you’re talking about area? Because how much area there is changes if the unit is small or big. If you don’t know the unit, you don’t really know what the area means. It’s just like with length. Twelve of a shorter unit is shorter than 12 of a longer unit. Part 2: Relate area to multiplication to draw rectangular arrays. T: S: T: S: T: Let’s draw a rectangular array with an area of 18 square centimeters. How might we find the side lengths? We could use our tiles to make the array and see. If you multiply side lengths you get area, so we can think about what numbers you can multiply to make 18. Work with your partner to make a list of multiplication facts that equal 18. (Possible list: 1 × 18, 18 × 1, 2 × 9, 9 × 2, 3 × 6, 6 × 3.) Let’s draw a 3 cm by 6 cm rectangular array. Use a ruler to measure the side lengths on your board. Draw hash marks for each centimeter and connect them to draw in all of the squares. After you’ve drawn your squares, check your work by skipcounting the rows to find the total number of tiles you drew. (Draw, label, and skip-count tiles in array.) Turn your board so that it’s vertical. Does the rectangle still have the same area? Yes. But the side lengths switched places! Tell your partner how you know the area is the same. The side lengths didn’t change, they just moved. It’s the commutative property. We learned before you can turn an array and it doesn’t change how much is in it; the rows just turn into columns and columns turn into rows. T: S: T: S: T: S: Lesson 7: Date: Interpret area models to form rectangular arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.28 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 3• 4 Part 3: Interpret area models to find area. T: The grid you drew inside of your 3 cm by 6 cm rectangle shows a picture of all the tiles that make up the area. Carefully erase the grid lines in your rectangle. (Pause.) The empty rectangle with labeled side lengths that’s left is called an area model. How can you find the total area just using the labeled side lengths? I can multiply! I can multiply the side lengths, 3 cm and 6 cm, to get the area, 18 square cm. (Project or draw the area model at right.) What is the total area of my 18 cm pictured rectangle? 18 square cm. Tell your partner how you figured out the area. It’s easy. One side length is 18 and the other is 1. 18 × 1 = 18. The labels tell you the unit is centimeters, so the area is square centimeters. 1 cm (Pass out the area model template.) Slip the area model template into your board. Use your ruler to measure the side lengths of one of the squares on the grid. (Allow students time to measure.) What unit is this grid made up of? Square inches! The side lengths of this area model aren’t labeled. Let’s draw a grid inside it to help us find the side lengths. Earlier we drew a grid inside a rectangle by drawing hash marks and using our ruler to connect the NOTES ON hash marks. Do we need to draw hash marks on the MULTIPLE MEANS area model to draw a grid inside it? FOR ACTION AND No, we can just use the grid lines. No, the lines on EXPRESSION: the grid can act as hash marks because the area model is lined up with the grid. Consider offering the following adaptations of the Problem Set: Use your ruler and the lines on the grid to draw Prompt students to approach squares inside the area model. (Allow students time to Rectangle E first. Offer practice with work.) What size are the units inside the area model? 1 by n rectangles to build fluency Square inches. They’re square inches because we and confidence. used the square-inch grid paper to help us draw the Remove side lengths to encourage squares. closer investigation. Find and label the side lengths, then write an equation Challenge students to devise an to find the area. alternate method to finding the area (Write 2 × 4 = 8 or 4 × 2 = 8.) of Benjamin’s bedroom floor. What is the area? 8 square inches! S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: Lesson 7: Date: Interpret area models to form rectangular arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.29 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 3• 4 Problem Set (10 minutes) Students should do their personal best to complete the Problem Set within the allotted 10 minutes. For some classes, it may be appropriate to modify the assignment by specifying which problems they work on first. Some problems do not specify a method for solving. Students solve these problems using the RDW approach used for Application Problems. Student Debrief (10 minutes) Lesson Objective: Interpret area models to form rectangular arrays. The Student Debrief is intended to invite reflection and active processing of the total lesson experience. Invite students to review their solutions for the Problem Set. They should check work by comparing answers with a partner before going over answers as a class. Look for misconceptions or misunderstandings that can be addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the lesson. You may choose to use any combination of the questions below to lead the discussion. What was your strategy for finding the total number of squares in Problem 2(c)? Invite students who drew arrays that demonstrate commutativity for Problem 4(a) (possibly 4 × 6 and 6 × 4) to share their work. Guide students to articulate understanding that commutativity still applies in the context of area. For Problem 4(b), most students answered that Mrs. Barnes’ array probably had 24 squares. Is there another answer that makes sense? (For example, 12, 48, 72.) Compare the area model to the array. How are they the same and different? (Guide discussion to include the commutativity of both models.) Lesson 7: Date: Interpret area models to form rectangular arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.30 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 3• 4 Exit Ticket (3 minutes) After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help you assess the students’ understanding of the concepts that were presented in the lesson today and plan more effectively for future lessons. You may read the questions aloud to the students. Lesson 7: Date: Interpret area models to form rectangular arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.31 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 Problem Set 3â€˘ 4 Name Date 1. Use a straight edge to draw a grid of equal size squares within the rectangle. Find and label the side lengths. Then multiply the side lengths to find the area. C A E D B F A. Area: _____ B. Area: _____ C. Area: _____ _____ = _____ square units _____ = _____ square units _____ = _____ square units D. Area: _____ E. Area: _____ F. Area: _____ _____ = _____ square units _____ = _____ square units _____ = _____ square units Lesson 7: Date: Interpret area models to form rectangular arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.32 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 Problem Set 3• 4 2. The area of Benjamin’s bedroom floor is shown on the grid to the right. Each = 1 square foot. How many total square feet is Benjamin’s floor? a. Label the side lengths. b. Use a straight edge to draw a grid of equal size squares within the rectangle. c. Find the total number of squares. Benjamin’s bedroom floor 3. Mrs. Young’s art class needs to create a mural that covers exactly 35 square feet. Mrs. Young marks the area for the mural as shown on the grid below. Each =1 square foot. Did she mark the area correctly? Explain your answer. Mural 4. Mrs. Barnes draws a rectangular array. Mila skip-counts by fours and Jorge skip-counts by sixes to find the total number of square units in the array. When they give their answers, Mrs. Barnes says that they are both right. a. Use pictures, numbers, and words to explain how Mila and Jorge can both be right. b. How many square units might Mrs. Barnes’ array have had? Lesson 7: Date: Interpret area models to form rectangular arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.33 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 Exit Ticket 3â€˘ 4 Name Date 1. Label the side lengths of Rectangle A on the grid below. Use a straight edge to draw a grid of equal size squares within Rectangle A. Find the total area of Rectangle A. Area: ________ square units Rectangle A 2. Mark makes a rectangle with 36 square-centimeter tiles. Gia makes a rectangle with 36 square-inch tiles. Whose rectangle has a bigger area? Explain your answer. Lesson 7: Date: Interpret area models to form rectangular arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.34 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 Homework 3â€˘ 4 Name Date 1. Find the area of each rectangular array. Label the side lengths of the matching area model and write a multiplication equation for each area model. Rectangular Arrays a. 3 _______ square units 2 3 _______ = _______ Area Models b. _______ square units _______ _______ = _______ c. _______ _______ = _______ _______ square units d. _______ _______ square units _______ = _______ Lesson 7: Date: Interpret area models to form rectangular arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.35 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 Homework 3• 4 3. Jillian arranges square pattern blocks into a 7 by 4 array. Draw Jillian’s array on the the grid below. How many square units are in Jillian’s rectangular array? a. b. Label the side lengths of Jillian’s array from Part (a) on the rectangle below. Then write a multiplication sentence to represent the area of the rectangle. 4. Fiona draws a 24 square-centimeter rectangle. Gregory draws a 24 square-inch rectangle. Whose rectangle is larger in area? How do you know? Lesson 7: Date: Interpret area models to form rectangular arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.36 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 Template 3• 4 Area Model Template Lesson 7: Date: Interpret area models to form rectangular arrays. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.37 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 3 Lesson 8 Objective: Find the area of a rectangle through multiplication of the side lengths. Suggested Lesson Structure Fluency Practice Application Problem Concept Development Student Debrief Total Time (11 minutes) (5 minutes) (34 minutes) (10 minutes) (60 minutes) Fluency Practice (11 minutes) Multiply by 6 3.OA.7 Group Counting 3.OA.1 (8 minutes) (3 minutes) Multiply by 6 (8 minutes) Materials: (S) Multiply by 6 Pattern Sheet (6–10) Note: This activity builds fluency with multiplication facts using units of 6. It works toward students knowing from memory all products of two one-digit numbers. See G3–M4–Lesson 2 for the directions for administration of a Multiply By pattern sheet. T: S: T: S: T: (Write 7 × 6 = ____.) Let’s skip-count up by sixes. (Count with fingers to 7 as students count.) 6, 12, 18, 24, 30, 36, 42. Let’s see how we can skip-count down to find the answer, too. (Show 10 fingers.) Start at 60. (Count down with your fingers as students say numbers.) 60, 54, 48, 42. (Distribute Multiply by 6 Pattern Sheet.) Let’s practice multiplying by 6. Be sure to work left to r ight across the page. Continue with the following possible sequence: 9 × 6, 6 × 6, and 8 × 6. Lesson 8: Date: Find the area of a rectangle through multiplication of the side lengths. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.38 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 3 Group Counting (3 minutes) Note: Group counting reviews interpreting multiplication as repeated addition. Direct students to count forward and backward, occasionally changing the direction of the count. Fours to 40 Sevens to 70 Eights to 80 Nines to 90 Application Problem (5 minutes) Marnie and Connor both skip-count square units to find the area of the same rectangle. Marnie counts, “3, 6, 9, 12, 15, 18, 21.” Connor counts, “7, 14, 21.” Draw what the rectangle might look like , then label the side lengths and find the area. Note: This problem reinforces G3–M4–Lesson 7 and sets the foundation for today’s Concept Development. Invite students to share their drawings and discuss how they are similar and how they are different. Concept Development (34 minutes) Materials: (S) Personal white board, inch ruler, grid template Part 1: Relate side lengths to area. T: S: T: S: (Project image shown to the right.) How many rows are in the incomplete array? 4 rows. How many square units are there in each row? 7 square units. Lesson 8: Date: Find the area of a rectangle through multiplication of the side lengths. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.39 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 3 T: S: T: S: T: MP.8 S: T: S: T: S: T: S: Talk to your partner: Do we need to complete the array to find the area of the rectangle? Why or why NOTES ON not? MULTIPLE MEANS Yes, then we can skip-count each row to find the total. OF ENGAGEMENT: No, we already know the side lengths! You may want to help English language How are the side lengths related to the area? learners relate the number of square If you multiply the side lengths together, the product is units in each row to the word columns, the same as the area. and relate columns and rows to side lengths. To some students it may Talk to a partner: Can you multiply any two side appear that these words are used lengths to find the area? interchangeably. Help clarify meaning. No, you have to multiply the side length that shows the number of rows times the side length that shows the number of squares in each row. What multiplication equation can be used to find the area of this rectangle? 4 × 7 = 28. In order to check our answer, use your grid template to trace and shade in an area model that is 4 units high and 7 units wide. Label each side length. (Draw and label.) Was our answer correct? Yes, I used the grid paper to count 28 squares inside. I skip-counted 4 sevens to make 28. Continue with the following possible sequence: 6 by 5, 8 by 7, and 9 by 6. Part 2: Use side lengths to find area. (Draw or project the rectangle shown at right.) T: S: T: S: T: S: What do you notice about this rectangle? We know the side lengths, but there is no grid inside. It’s an area model. Do we still have enough information to find the area of this rectangle, even without the grid lines inside? Yes! We know both side lengths. Write the multiplication equation to find the area of this rectangle. 6 × 8 = 48. 6 cm Area = ? sq cm 8 cm NOTES ON MULTIPLE MEANS OF ENGAGEMENT: You may choose to have students work through these examples independently or in pairs. Lesson 8: Date: Find the area of a rectangle through multiplication of the side lengths. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.40 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 3 Continue with the following suggested examples: 9 cm 5 cm 9 in 7 ft Area = ? sq cm 8 in Area = ? sq in 10 ft Area = ? sq ft Part 3: Use area and side length to find unknown side length. (Draw or project the rectangle shown at right.) T: S: T: What do you notice about this rectangle? We know the area, but not both side lengths. One of the side lengths is unknown. Write a multiplication equation on your board to show how to find the area of this rectangle. Use a question mark for the unknown side length. (Write 3 × ? = 27.) What is the value of the question mark? 9! How do you know? I know that 3 times 9 equals 27! So what is the unknown side length? 9 centimeters! Write the related division equation on your board. (Write 27 3 = 9.) ? cm 3 cm Area = 27 sq cm S: T: S: T: S: T: S: T: S: Continue with the following suggested examples: ? ft ? cm 4 cm Area = 32 sq cm 8 ft 7 in Area = 24 sq ft ? in Area = 42 sq in T: S: When you know the area and one side length of a rectangle, how can you find the other side length? I can think of it as a multiplication equation with a missing factor. Or, I can divide the area by the known side length. Lesson 8: Date: Find the area of a rectangle through multiplication of the side lengths. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.41 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 3 Problem Set (10 minutes) Students should do their personal best to complete the Problem Set within the allotted 10 minutes. For some classes, it may be appropriate to modify the assignment by specifying which problems they work on first. Some problems do not specify a method for solving. Students solve these problems using the RDW approach used for Application Problems. Student Debrief (10 minutes) Lesson Objective: Find the area of a rectangle through multiplication of the side lengths. The Student Debrief is intended to invite reflection and active processing of the total lesson experience. Invite students to review their solutions for the Problem Set. They should check work by comparing answers with a partner before going over answers as a class. Look for misconceptions or misunderstandings that can be addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the lesson. You may choose to use any combination of the questions below to lead the discussion. In what way is the area of Problem 1(b) related to the area of Problem 1(a)? (It is double.) How could you use the side lengths to help you figure out that 8 × 7 is double 4 × 7? Explain how you can tell a shape is a square just by looking at the side lengths (Problem 1(c)). How are the rectangles in Problem 1(a) and 2(c) similar? How are they different? Address the following possible misconception in Problem 4. Even though Eliza’s bedroom has 1 side length (6 feet) that is 1 more than her brother’s bedroom (5 feet), and 1 side length (7 feet) that is 1 less than her brother’s bedroom (8 feet), the floor areas are not equal. Why is there a connection between a rectangle’s side lengths and its area? Lesson 8: Date: Find the area of a rectangle through multiplication of the side lengths. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.42 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 3 Exit Ticket (3 minutes) After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help you assess the studentsâ€™ understanding of the concepts that were presented in the lesson today and plan more effectively for future lessons. You may read the questions aloud to the students. Lesson 8: Date: Find the area of a rectangle through multiplication of the side lengths. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.43 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 Pattern Sheet 3 Multiply. Lesson 8: Date: Find the area of a rectangle through multiplication of the side lengths. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.44 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 Problem Set 3 Name 1. Write a multiplication sentence to find the area of each rectangle. a. b. Date 7 ft Area: ______ sq ft 7 ft c. 6 ft 4 ft 8 ft Area: ______ sq ft 6 ft Area: ______ sq ft _______ × _______ = _______ . _______ × _______ = _______ 2. Write a multiplication sentence and a division sentence to find the unknown side length for each rectangle. _____ ft _____ ft 4 ft c. b. a. 3 ft Area = 15 sq ft 9 ft Area = 72 sq ft _____ ft Area = 28 sq ft _______ × _______ = _______ _______ _______ = _______ _______ × _______ = _______ _______ × _______ = _______ _______ _______ = _______ _______ _______ = _______ _______ × _______ = _______ 3. On the grid below, draw a rectangle that has an area of 42 square inches. Label the side lengths. Lesson 8: Date: Find the area of a rectangle through multiplication of the side lengths. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.45 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 Problem Set 3 4. Ursa draws a rectangle that has side lengths of 9 centimeters and 6 centimeters. What is the area of the rectangle? Explain how you found your answer. 5. Eliza’s bedroom measures 6 feet by 7 feet. Her brother’s bedroom measures 5 feet by 8 feet. Eliza says their rooms have the same exact floor area. Is she right? Why or why not? 6. Cliff draws a rectangle with a side length of 6 inches and an area of 24 square inches. What is the other side length? How do you know? Lesson 8: Date: Find the area of a rectangle through multiplication of the side lengths. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.46 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 Exit Ticket 3 Name Date 1. Write a multiplication sentence to find the area of the rectangle below. 9 inches 3 inches Area: _____ sq in _______ × _______ = _______ 2. Write a multiplication sentence and a division sentence to find the unknown side length for the rectangle below. _____ inches 6 inches Area: 54 sq in _______ × _______ = _______ _______ × _______ = _______ _______ _______ = _______ Lesson 8: Date: Find the area of a rectangle through multiplication of the side lengths. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.47 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 Homework 3 Name 1. Write a multiplication sentence to find the area of each rectangle. a. 8 cm 3 cm Area: ______ sq cm b. Date 8 cm 6 cm Area: ______ sq cm _______ × _______ = _______ _______ × _______ = _______ c. 4 ft d. 7 ft 4 ft Area: ______ sq ft 4 ft Area: ______ sq ft _______ × _______ = _______ _______ × _______ = _______ 2. Write a multiplication sentence and a division sentence to find the unknown side length for each rectangle. a. 9 ft _____ ft. b. 3 ft Area: 24 sq ft _____ ft Area: 36 sq ft _______ × _______ = _______ _______ _______ = _______ Lesson 8: Date: _______ × _______ = _______ _______ _______ = _______ 4.B.48 Find the area of a rectangle through multiplication of the side lengths. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 Homework 3 2. On the grid below draw a rectangle that has an area of 32 square centimeters. Label the side lengths. 3. Patricia draws a rectangle that has side lengths of 4 centimeters and 9 centimeters. What is the area of the rectangle? Explain how you found your answer. 4. Charles draws a rectangle with a side length of 9 inches and an area of 27 square inches. What is the other side length? How do you know? Lesson 8: Date: Find the area of a rectangle through multiplication of the side lengths. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.49 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 Template 3 Lesson 8: Date: Find the area of a rectangle through multiplication of the side lengths. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.B.50 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org New York State Common Core GRADE 3 Mathematics Curriculum GRADE 3 • MODULE 4 Topic C Arithmetic Properties Using Area Models 3.MD.5, 3.MD.6, 3.MD.7 Focus Standard: 3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). Relate area to the operations of multiplication and addition. a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent wholenumber products as rectangular areas in mathematical reasoning. c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning. d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems. Addition and Subtraction of Length Units Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 Multiplication and Division with Units of 0, 1, 6–9, and Multiples of 10 Multi-Digit Multiplication and Division Exploring Multiplication 3.MD.6 3.MD.7 Instructional Days: Coherence -Links from: 3 G2–M2 G3–M1 G3–M3 -Links to: G4–M3 G4–M7 Topic C begins with a concrete study of arithmetic properties. Students cut apart rectangular grids and rearrange the parts to create new rectangles with the same area. Lesson 9 lays the foundation for the work to come in Lessons 10 and 11. Topic C: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Arithmetic Properties Using Area Models 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported.License. 4.C.1 NYS COMMON CORE MATHEMATICS CURRICULUM Topic C 3â€˘ 4 In Lesson 10, students apply knowledge of the distributive property from Modules 1 and 3 to find area. In previous modules, they learned to decompose an array of discrete items into two parts, determine the number of units in each part, and then find the sum of the parts. Now students connect this experience to using the distributive property to determine the missing side length of an array that may, for example, have an area of 72 square units. They might decompose the area into an 8 by 5 rectangle and an 8 by 4 rectangle. The sum of the side lengths, 5 + 4, gives them the length of the missing side. In Lesson 11, students use a given number of square units to determine all possible whole number side lengths of rectangles with that area. They engage in MP.3 as they justify that they have found all possible solutions for each given area using the associative property. Areas of 24, 36, 48, and 72 are chosen to reinforce multiplication facts that are often more difficult. Students realize that different factors give the same product. For example, they find that 4 by 12, 6 by 8, 1 by 48, and 2 by 24 arrays all have an area of 48 square units. They use understanding of the commutative property to recognize that area models can be rotated similar to the arrays in Modules 1 and 3. A Teaching Sequence Towards Mastery of Arithmetic Properties Using Area Models Objective 1: Analyze different rectangles and reason about their area. (Lesson 9) Objective 2: Apply the distributive property as a strategy to find the total area of a larger rectangle by adding two products. (Lesson 10) Objective 3: Demonstrate the possible whole number side lengths of rectangles with areas of 24, 36, 48, or 72 square units using the associative property. (Lesson 11) Topic C: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Arithmetic Properties Using Area Models 10/1/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported.License. 4.C.2 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 3• 4 Lesson 9 Objective: Analyze different rectangles and reason about their area. Suggested Lesson Structure Fluency Practice Application Problems Concept Development Student Debrief Total Time (12 minutes) (5 minutes) (33 minutes) (10 minutes) (60 minutes) Fluency Practice (12 minutes) Group Counting 3.OA.1 Find the Area 3.MD.6 Decompose the Multiplication Sentence 3.OA.5 (4 minutes) (4 minutes) (4 minutes) Group Counting (4 minutes) Note: Group counting reviews interpreting multiplication as repeated addition. Direct students to count forward and backward, occasionally changing the direction of the count. Fours to 40 Sevens to 70 Eights to 80Nines to 90 Find the Area (4 minutes) Note: This fluency reviews strategies for finding the area of a rectangle. T: S: T: S: T: S: T: S: (Project a rectangular array with 2 rows of 4 units. Write 1 tile = 1 square meter.) What does 1 tile equal? 1 square meter. (Point to the side length of 4 units.) This is the length of the rectangle. What is its value? 4 meters. (Point to the side length of 2 units.) This is the width of the rectangle. What is its value? 2 meters. Write a multiplication sentence to represent the area of the rectangle. (Write 2 m × 4 m = 8 sq m or 4 m × 2 m = 8 sq m.) Lesson 9: Date: Analyze different rectangles and reason about their area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.3 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 3• 4 Continue with the following possible sequence: 3 rows of 5 units, 3 rows of 7 units, 4 rows of 6 units, 4 rows of 9 units, and 6 rows of 8 units. Decompose the Multiplication Sentence (4 minutes) Materials: (S) Personal white boards Note: This activity anticipates the distributive property used in G3 –M4–Lesson 10, while reviewing G3– Module 3 concepts. T: S: T: S: T: S: (Write 8 × 6 = (5 + ) × 6.) On your boards, complete 8 × 6 = (5 + 3) × 6 the number sentence. = (5 × 6) + (3 × 6) (Write 8 × 6 = (5 + 3) × 6.) = 30 + 18 (Write = ( × 6) + ( × 6).) Complete the number = 48 sentence. (Write (5 × 6) + (3 × 6).) Solve the multiplication problems and write an addition sentence. Below it, write your answer. (Write 30 + 18 and 48 below it.) Continue with the following possible sequence: 7 × 6, 6 × 6, and 9 × 6. Application Problem (5 minutes) Mario plans to completely cover his 8-inch by 6-inch cardboard with square-inch tiles. He has 42 square-inch tiles. How many more square-inch tiles does Mario need to cover the cardboard without any gaps or overlap? Explain your answer. Note: This problem reviews the concept of finding area. Students will likely solve by multiplying side lengths (shown above), having just practiced this strategy in G3–M4–Lesson 8. Lesson 9: Date: Analyze different rectangles and reason about their area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.4 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 3• 4 Concept Development (33 minutes) Materials: (S) Centimeter grid, personal white boards, Problem Set Problems 1 and 2 in the Problem Set: T: S: How can we cut this grid to get 2 equal rectangles? Cut it in half. If we cut on the line between the fifth and sixth squares, we’ll have 2 equal rectangles. If we fold the grid in half and cut along the fold, we can make 2 equal rectangles. Do that now, and then answer Problem 1(a). T: NOTES ON MULTIPLE MEANS OF ACTION AND EXPRESSION: Cutting paper with scissors may be a challenge for some learners. Precision is important to this lesson. Please try the following tips: Provide centimeter grids on cardstock or thicker paper. T: S: T: S: T: S: T: S: T: S: T: How can you find the area of one of the rectangles? Darken and thicken the cutting Multiply the side lengths. Multiply 5 times 10. lines. Provide left-handed, loop, spring, Answer Problem 1(b). (Allow students time to work.) self-opening, or other adaptive What is the area of one of the rectangles? scissors, if needed. 50 square centimeters! Instruct students to turn the paper, What is the area of the other rectangle? How do you not the scissors. know? Offer precut centimeter grids. 50 square centimeters because the rectangles are equal. How can you find the total area of the rectangles? Add 50 square centimeters plus 50 square centimeters. Answer Problem 1(c). (Allow students time to work.) What is the total area? 100 square centimeters. Place your rectangles next to each other to make 1 long rectangle. Talk to a partner. What do you think the area of this long rectangle is? Why? S: 100 square centimeters because I added 50 square centimeters plus 50 square centimeters. 100 square centimeters because that’s the total area of the smaller rectangles and that doesn’t change when we move them to make the longer rectangle. Lesson 9: Date: Analyze different rectangles and reason about their area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.5 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 3• 4 T: S: T: S: T: S: T: S: MP.3 Let’s see if you are right! Answer Problem 2(a). (Allow students time to work.) What multiplication fact can help you find the area of this longer rectangle? 5 × 20. How can you solve this multiplication fact? We can think of it as 5 times 2 tens. We could think of it as 5 (2 10), which is the same as (5 × 2) × 10. We can think of it the same way as before, as 2 equal rectangles. Choose a strategy and use it to answer Problem 2(b). (Allow students time to work.) What is the area of this longer rectangle? 100 square centimeters! Was your prediction about the area of this longer rectangle correct? Yes! Repeat this process, instructing students to fold 2 columns behind one of the rectangles, so they now have a 5 by 8 rectangle and a 5 by 10 rectangle. They can use their boards to record the total area of the 2 separate rectangles and the area of the longer rectangle that is made by joining the 2 smaller rectangles. T: S: T: S: What did you notice about the sum of the areas of the 2 small rectangles and the area of the longer rectangle? They’re the same! How can we use the areas of 2 small rectangles that form a longer rectangle to find the area of the longer rectangle? Add the areas of the smaller rectangles! Problem Set (10 minutes) Students should do their personal best to complete the Problem Set within the allotted 10 minutes. For some classes, it may be appropriate to modify the assignment by specifying which problems they work on first. Some problems do not specify a method for solving. Students solve these problems using the RDW approach used for Application Problems. Student Debrief (10 minutes) Lesson Objective: Analyze different rectangles and reason about their area. The Student Debrief is intended to invite reflection and active processing of the total lesson experience. Invite students to review their solutions for the Problem Set. They should check work by comparing answers with a partner before going over answers as a class. Look for misconceptions or misunderstandings that can be Lesson 9: Date: Analyze different rectangles and reason about their area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.6 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 3• 4 addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the lesson. You may choose to use any combination of the questions below to lead the discussion. Talk to a partner: In Problem 1(a) how does knowing the side lengths of the grid help you find the side lengths of the small rectangles without counting? Did anyone use the break apart and distribute strategy to solve Problem 2(b)? Explain what you broke apart. Why did you make that choice? (In anticipation of G3–M4–Lesson 10, which uses the distributive property, ask students how the paper rectangles show the distributive property.) Compare the equations you used to solve Problems 1(b) and 2(b). How are they the same? How are they different? Explain to a partner how you found the length and width for the new rectangle in Problem 3(b). Did anyone multiply the side lengths to solve Problem 3(c)? What strategy did you use to multiply 4 × 13? How was Problem 4 different from the rest of the problems? Exit Ticket (3 minutes) After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help you assess the students’ understanding of the concepts that were presented in the lesson today and plan more effectively for future lessons. You may read the questions aloud to the students. Lesson 9: Date: Analyze different rectangles and reason about their area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.7 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 Problem Set 3â€˘ 4 Name 1. Cut the grid into 2 equal rectangles. a. Draw and label the side lengths of the 2 rectangles. Date b. Write an equation to find the area of 1 of the rectangles. c. Write an equation to show the total area of the 2 rectangles. 2. Place your 2 equal rectangles side by side to create a new, longer rectangle. a. Draw an area model to show the new rectangle. Label the side lengths. b. Find the total area of the longer rectangle. Lesson 9: Date: Analyze different rectangles and reason about their area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.8 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 Problem Set 3• 4 3. Furaha and Rahema use square tiles to make the rectangles shown below. Furaha’s Rectangle Rahema’s Rectangle a. Label the side lengths on the rectangles above and find the area of each rectangle. b. Furaha pushes his rectangle next to Rahema’s rectangle to form a new, longer rectangle. Draw an area model to show the new rectangle. Label the side lengths. c. Rahema says the area of the new, longer rectangle is 52 square units. Is she right? Explain your answer. 4. Kiera says she can find the area of the long rectangle below by adding the areas of Rectangles A and B. Is she right? Why or why not? Rectangle A Rectangle B Lesson 9: Date: Analyze different rectangles and reason about their area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.9 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 Exit Ticket 3â€˘ 4 Name Lamar uses square tiles to make the 2 rectangles shown below. Date Rectangle A Rectangle B a. Label the side lengths of the 2 rectangles. b. Write equations to find the areas of the rectangles. Area of Rectangle A: _______________ Area of Rectangle B: _______________ c. Lamar pushes Rectangle A next to Rectangle B to make a bigger rectangle. What is the area of the bigger rectangle? How do you know? Lesson 9: Date: Analyze different rectangles and reason about their area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.10 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 Homework 3â€˘ 4 Name 1. Use the grid to answer the questions below. Date a. Draw a line to show how to divide the grid into 2 equal rectangles. Shade in 1 of the rectangles. b. Label the side lengths of each rectangle. c. Write an equation to show the total area of the 2 rectangles. Lesson 9: Date: Analyze different rectangles and reason about their area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.11 ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 Homework 3• 4 2. Alexa cuts out the 2 equal rectangles from Problem 1(a) and puts the two shorter sides together. a. Draw Alexa’s new rectangle and label the side lengths below. b. Find the total area of the new, longer rectangle. c. Is the area of the new, longer rectangle equal to the total area in Problem 1(c)? Explain why or why not. Lesson 9: Date: Analyze different rectangles and reason about their area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.12 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 Template 3• 4 Lesson 9: Date: Analyze different rectangles and reason about their area. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.13 © 2013 Common Core, Inc. Some rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 3 Lesson 10 Objective: Apply the distributive property as a strategy to find the total area of a large rectangle by adding two products. Suggested Lesson Structure Fluency Practice Application Problem Concept Development Student Debrief Total Time (8 minutes) (5 minutes) (37 minutes) (10 minutes) (60 minutes) Fluency Practice (8 minutes) Group Counting 3.OA.1 Find the Unknown Factor 3.OA.4 (3 minutes) (5 minutes) Group Counting (3 minutes) Note: Group counting reviews interpreting multiplication as repeated addition. Direct students to count forward and backward, occasionally changing the direction of the count. Sixes to 60 Sevens to 70 Eights to 80 Nines to 90 Find the Unknown Factor (5 minutes) Materials: (S) Personal white boards Note: This fluency anticipates finding all possible side lengths of rectangles with areas of 12, 24, 36, 48, and 72 square units in G3–M4–Lesson 11. T: S: T: S: (Write 4 × 4 × 3 = 12. = 12.) Fill in the unknown factor to make a true number sentence. = 12, 2 × = 12, and 3 × = 12. Continue with the following possible sequence: 6 × (Write 8 × = 24.) Fill in the unknown factor to make a true number sentence. (Write 8 × 3 = 24.) Lesson 10: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Apply the distributive property as a strategy to find the total area of a large rectangle by adding two products. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.14 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 3 Continue with the following possible sequence: 6× 4× 2× 12 × 3× = 24, 3 × = 36, 8 × = 24, 12 × = 72, 3 × = 72. = 24, 6 × = 36, 9 × = 36, 4 × = 24, = 72, 8 × = 48, 9 × = 72, 6 × = 48, = 24, 12 × = 36, 12 × = 48, = 36, 4 × = 48, 6 × = 72, and A NOTE ON 12 AS A FACTOR: The suggested sequence for this fluency activity leads students to solve number sentences with 12 as a factor. While some students might be fluent with these facts, others might rely on the distributive property to write true number sentences. The expectation is for students to become familiar with 12 as a factor, since these number sentences will be seen in G3–M4– Lesson 11. Application Problem (5 minutes) Sonya folds a 6 by 6 square inch piece of paper into 4 equal parts, shown below. What is the area of 1 of the parts? Note: This problem reviews the concept of finding area. Concept Development (37 minutes) Materials: (S) Personal white boards, square-centimeter tiles, tiling template Students start with the tiling template in their personal white boards. T: (Project the tiling template.) There are 3 rectangles we are going to focus on: the large rectangle (trace the outside of the large rectangle with your finger), the shaded rectangle (trace the shaded rectangle), and the unshaded rectangle (trace the unshaded rectangle). 6 Use square-centimeter tiles to find the area of the large rectangle. (Allow students time to work.) What is the area of the large rectangle? 5 5×6 48 square centimeters! Use square-centimeter tiles to find the side lengths of the shaded rectangle. (Allow students time to work.) 3 What are the side lengths? 3×6 5 centimeters and 6 centimeters! (5 × 6) + (3 × 6) Label the side lengths. (Allow students time to label the = 30 + 18 side lengths.) What multiplication expression can you = 48 square units Lesson 10: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org T: S: T: S: T: Apply the distributive property as a strategy to find the total area of a large rectangle by adding two products. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.15 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 3 S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: MP.7 use to find the area of the shaded rectangle? 5 × 6! Write that expression next to the shaded rectangle. (Allow students time to write expression.) What side length do we already know for the unshaded rectangle? 6 centimeters! Use square-centimeter tiles to find the other side length of the unshaded rectangle. (Allow students time to work.) What is the other side length? 3 centimeters! Label the side length. (Allow students time to label the side length.) What multiplication expression can you use to find the area of the unshaded rectangle? 3 × 6! Write that expression next to the unshaded rectangle. (Allow students time to write expression.) How can we use these two expressions to help us find the area of the large rectangle? We can add them! The area of the shaded rectangle plus the area of the unshaded rectangle equals the area of the large rectangle. Write an expression on your board to show this. (Write (5 × 6) + (3 × 6).) Read your expression to a partner, and then find its value. (Allow students time to solve.) What is the area of the large rectangle? 48 square units! Is that the answer you got when you tiled the large rectangle? Yes! Write the value of the length of the large rectangle as an addition expression. (Write 5 + 3.) What will you multiply by to find the area? 6! Write that in your expression. Where should we put parentheses? Around 5 + 3, because we need to add first to find the side length, then we can multiply. Add the parentheses to your expression. What is 5 + 3? 8! What is the new expression? 8 × 6. What is the area? 48 square units! 8×6 Is that the same answer we just got? Yes! (5 + 3) × 6 (Write the expressions as shown.) How are these three (5 × 6) + (3 × 6) expressions related? They all show the area of the large rectangle. Oh Lesson 10: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Apply the distributive property as a strategy to find the total area of a large rectangle by adding two products. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.16 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 3 MP.7 T: S: look, they show the break apart and distribute strategy! Yeah, they show that the side length 8 is broken apart into 5 plus 3. Then 5 and 3 are multiplied by the other side length, 6. Discuss with a partner how the large rectangle on your board also shows the break apart and distribute strategy. (Discuss.) NOTES ON MULTIPLE MEANS OF ACTION AND EXPRESSION: Repeat the process with the following possible suggestions, providing pictures of rectangles with grid lines: T: S: T: S: A 15 by 8 rectangle. (Students can partition as (10 + 5) × 8. This will help students see that this Consider directing students who may strategy is helpful when they cannot multiply the not complete the Problem Set within side lengths because they do not know these facts.) the allotted time to Problem 2 for An 18 by 9 rectangle. (Students can decompose as valuable application and double 9 × 9 or (10 + 8) × 9.) demonstration of understanding of We broke apart the 18 by 9 rectangle into two 9 by 9 today’s objective. Offer planning and strategy development support to rectangles. What other ways could we break apart this learners, if needed. Model a thinkrectangle? aloud in which you consider two or I would do 10 by 9 and 8 by 9 rectangles. more possibilities, reason about your Explain to a partner the process you use to decide how selection, and solve. to break apart a side length. I look for facts I know. I try to find a way to make a 5 or 10 because they’re easy facts. Problem Set (10 minutes) Students should do their personal best to complete the Problem Set within the allotted 10 minutes. For some classes, it may be appropriate to modify the assignment by specifying which problems they work on first. Some problems do not specify a method for solving. Students solve these problems using the RDW approach used for Application Problems. Student Debrief (10 minutes) Lesson Objective: Apply the distributive property as a strategy to find the total area of a large rectangle by adding two products. The Student Debrief is intended to invite reflection and active processing of the total lesson experience. Invite students to review their solutions for the Problem Set. They should check work by comparing answers with a Lesson 10: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Apply the distributive property as a strategy to find the total area of a large rectangle by adding two products. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.17 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 3 partner before going over answers as a class. Look for misconceptions or misunderstandings that can be addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the lesson. You may choose to use any combination of the questions below to lead the discussion. What are the side lengths of the large rectangle in Problem 1(c)? Can you multiply these side lengths to find the area? How does the break apart and distribute strategy help you? Without multiplying the side lengths of the large rectangle in Problem 1(d), how could you check to make sure your answer is right? (Students might say count the squares or skip-count by eight 12 times.) Discuss with a partner, which strategy is most efficient, counting squares, skipcounting, or using the break apart and distribute strategy? How was setting up and solving Problem 2 different from the rest of the problems? What side length did you break apart in Problem 3, and how did you break it apart? Why? With a partner, list as many possibilities as you can for how you could use the break apart and distribute strategy to find the area of a rectangle with side lengths of 20 and 7. Can we break it into 3 parts if we want to? Which one of your possibilities would you use to solve this problem? Why? Exit Ticket (3 minutes) After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help you assess the students’ understanding of the concepts that were presented in the lesson today and plan more effectively for future lessons. You may read the questions aloud to the students. Lesson 10: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Apply the distributive property as a strategy to find the total area of a large rectangle by adding two products. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.18 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 Problem Set 3 Name Date 1. Label the side lengths of the shaded and unshaded rectangles. Then find the total area of the large rectangle by adding the areas of the two smaller rectangles. a. 7 b. 4 5 12 × 4 = ( ______ + 2) × 4 = ( ______ × 4) + ( 2 × 4) = ______ + 8 3 = _____ square units 8 × 7 = (5 + 3) × 7 = (5 × 7) + (3 × 7) = ______ + _____ = ______ square units 2 c. d. 6 6 × 13 = 6 × (________ + 3) = (6 ×______) + (6 × 3) = ______ + ______ = ______ square units 8 × 12 = 8 × ( ____ + _____ ) = (8 × ____ ) + (8 × ____ ) = ______ + ______ = ______ square units Lesson 10: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Apply the distributive property as a strategy to find the total area of a large rectangle by adding two products. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.19 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 Problem Set 3 2. Vince imagines 1 more row of eight to find the total area of a 9 × 8 rectangle. Explain how this could help him solve 9 × 8. 3. Shade to break the 15 × 5 rectangle into 2 smaller rectangles. Then find the sum of the areas of the 2 smaller rectangles to find the total area. Explain your thinking. Lesson 10: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Apply the distributive property as a strategy to find the total area of a large rectangle by adding two products. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.20 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 Exit Ticket 3 Name Date Label the side lengths of the shaded and unshaded rectangles. Then find the total area of the large rectangle by adding the areas of the 2 smaller rectangles. a. b. 8 × 7 = 8 × ( _____ + _____ ) = (8 ×_____ ) + (8 ×_____ ) = ______ + ______ = ______ square units 9 × 13 = 9 × ( _____ + _____ ) = ( ____ × _____ ) + ( _____×_____ ) = ______ + ______ = ______ square units Lesson 10: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Apply the distributive property as a strategy to find the total area of a large rectangle by adding two products. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.21 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 Homework 3 Name Date 1. Label the side lengths of the shaded and unshaded rectangles. Then find the total area of the large rectangle by adding the areas of the 2 smaller rectangles. a. 8 b. 5 5 12 × 5 = ( ______ + 2) × 5 4 = ( ______ × 5) + ( 2 × 5) = ______ + 10 = ______square units 9 × 8 = (5 + 4) × 8 = (5 × 8) + (4 × 8) = ______ + _____ = ______ square units 2 c. d. 7 7 × 13 = 7 × ( _______ + 3) = (7 × _____) + (7 × 3) = ______ + ______ 9 × 12 = 9 × ( _____ + _____ ) = (9 × _____ ) + (9 × _____ ) = ______ + ______ = ______ square units = ______ square units Lesson 10: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Apply the distributive property as a strategy to find the total area of a large rectangle by adding two products. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.22 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 Homework 3 2. Finn imagines 1 more row of nine to find the total area of 9 × 9 rectangle. Explain how this could help him solve 9 × 9. 3. Shade to break the 16 × 4 rectangle into 2 smaller rectangles. Then find the sum of the areas of the 2 smaller rectangles to find the total area. Explain your thinking. Lesson 10: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Apply the distributive property as a strategy to find the total area of a large rectangle by adding two products. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.23 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 Template 3 Lesson 10: Date: ÂŠ 2013 Common Core, Inc. Some rights reserved. commoncore.org Apply the distributive property as a strategy to find the total area of a large rectangle by adding two products. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.24 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 3• 4 Lesson 11 Objective: Demonstrate possible whole number side lengths of rectangles with areas of 24, 36, 48, or 72 square units using the associative property. Suggested Lesson Structure Fluency Practice Application Problem Concept Development Student Debrief Total Time (13 minutes) (5 minutes) (32 minutes) (10 minutes) (60 minutes) Fluency Practice (13 minutes) Group Counting 3.OA.1 Find the Unknown Factor 3.OA.4 Find the Area 3.MD.7 (3 minutes) (5 minutes) (5 minutes) Group Counting (3 minutes) Note: Group counting reviews interpreting multiplication as repeated addition. Direct students to count forward and backward, occasionally changing the direction of the count. Sixes to 60 Sevens to 70 Eights to 80 Nines to 90 Find the Unknown Factor (5 minutes) Materials: (S) Personal white boards Note: This fluency anticipates the objective of today’s lesson . T: S: T: S: (Write 6 × 6 × 2 = 12. = 12.) Fill in the unknown factor to make a true number sentence. = 12, 2 × = 12, and 3 × = 12. Continue with the following possible sequence: 4 × (Write 3 × = 24.) Fill in the unknown factor to make a true number sentence. (Write 3 × 8 = 24.) Lesson 11: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Demonstrate possible whole number side lengths of rectangles with areas of 24, 36, 48, or 72 square units using the associative property. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.25 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 3• 4 Continue with the following possible sequence: 4 × 9× = 36, 9 × = 72, 6 × = 48, 8 × = 72, 8 × = 24, 8 × = 24, 6 × = 36, 4 × = 48, and 2 × = 24. = 36, 6 × = 24, Find the Area (5 minutes) Materials: (S) Personal white boards Note: This fluency reviews using the distributive property from G3–M4–Lesson 10. T: (Project the rectangle as shown.) On your boards, write an expression that we could use to find the area of the shaded rectangle. (Write 3 × 5.) On your boards, write an expression that we could use to find the area of the unshaded rectangle. (Write 3 × 3.) How can you use these expressions to find the area of the large rectangle? Add them! Write an equation, showing the sum of the shaded and unshaded rectangles. Below it, write the area of the entire rectangle. (Write 15 + 9 = 24 square units.) 3 5 3 S: T: S: T: S: T: (3 × 5) + (3 × 3) = 15 + 9 = 24 square units S: Continue with the following possible sequence: 9 × 5 = (5 × 5) + (4 × 5), 13 × 4 = (10 × 4) + (3 × 4), and 17 × 3 = (10 × 3) + (7 × 3). NOTES ON MULTIPLE MEANS OF ENGAGEMENT: Alternatively, challenge students working above grade level with this length unknown version: One fourth of the banquet table has an area of 9 square feet. If the width of the table is 3 feet, what is the length? What is the area of the table? Application Problem (5 minutes) The restaurant’s banquet table measures 3 feet by 6 feet. For a large party, workers at the restaurant place 2 banquet tables side by side to create 1 long table. Find the area of the new, longer table. Lesson 11: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Demonstrate possible whole number side lengths of rectangles with areas of 24, 36, 48, or 72 square units using the associative property. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.26 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 3• 4 Note: This problem reviews G3–M4–Lesson 10’s concept of applying the distributive property to find the total area of a large rectangle by adding two products. It also reviews factors of 36 and multiples of 12 that lead into the Concept Development. Concept Development (32 minutes) Materials: (S) Personal white boards T: S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: T: S: 12 Write an expression to show how to find the area of a rectangle with side lengths 3 and 12. 3 (Write 3 × 12.) In the Application Problem, you found that 3 times 12 is? 36! So, the area of this rectangle is? 36 square units! (Write 3 × (2 × 6).) Is this expression equal to the one you just wrote? Yes, you just wrote 12 as 2 × 6. Write this expression on your board with the parentheses in a different place. At my signal, show me your board. (Signal.) (Show (3 × 2) × 6.) Solve 3 × 2 and write the new expression on your board. (Allow students time to work.) Whisper the new expression to a partner. 6 × 6. What new side lengths did we find for a rectangle with an area of 36 square units? 6 and 6! Let’s look at our expression, (3 × 2) × 6, again. Use the commutative property and switch the order of the factors in the parentheses. (Write (2 × 3) × 6.) Will you be able to find new side lengths by moving the parentheses? (Write 2 × (3 × 6).) Yes, it’ll be 2 and 18! (Write 3 × (3 × 4).) Is this expression equal to our first one, 3 × 12? Yes, now you wrote 12 as 3 × 4. Write this expression on your board with the parentheses in a different place. At my signal, show me your board. (Signal.) (Show (3 × 3) × 4.) Solve 3 × 3 and write the expression on your board. (Allow students time to work.) Whisper the new expression to a partner. 9 × 4. Lesson 11: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Demonstrate possible whole number side lengths of rectangles with areas of 24, 36, 48, or 72 square units using the associative property. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.27 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 3• 4 T: S: T: S: T: S: T: S: T: S: T: S: T: MP.3 S: T: S: T: S: T: S: What new side lengths did we find for a rectangle with an area of 36 square units? 9 and 4! Let’s look at our expression, (3 × 3) × 4 again. If I use the commutative property and switch the order of the factors in the parentheses, will I be able to find new side lengths by moving the parentheses? No, it’ll still be 9 and 4. No, because both factors in the parentheses are 3, so switching their order won’t change the numbers you get when you move the parentheses. Do you think we found all the possible whole number side lengths for this rectangle? Yes. I’m not sure. Let’s look at our side lengths. Do you have a side length of 1? No! We forgot the easiest one. It’s 1 and 36! Do we have a side length of 2? Yes. 3? Yes. Work with a partner to look at the rest of your side lengths to see if you have the numbers 4 through 10. (Allow students time to work.) Which of these numbers, 4 through 10, aren’t included in your side lengths? 5, 7, 8, and 10. Discuss with a partner why these numbers aren’t in your list of side lengths. 5, 7, 8, and 10 can’t be side lengths because there aren’t any whole numbers we can multiply these numbers by to get 36. Would any two-digit times two-digit number work? No, they would be too big. No, because we know 10 × 10 equals 100 and that’s bigger than 36. NOTES ON Now do you think we found all the possible side whole MULTIPLE MEANS OF number side lengths for a rectangle with an area of 36 REPRESENTATION: square units? Extend Problem 1 for students working Yes! above grade level by inviting experimentation and choice in placing parentheses, as well as number order, in the multiplication sentences. For example, ask, “What would happen if we changed it to 4 × 6 × 2?” Encourage students to discuss or journal about their discoveries. Assist English language learners by rephrasing Problem 4 in multiple ways. You might ask, “How does the difference between the length and width of the rectangle change the shape?” Repeat the process with rectangles that have areas of 24, 48, and 72 square units. Problem Set (10 minutes) Students should do their personal best to complete the Problem Set within the allotted 10 minutes. For some classes, it may be appropriate to modify the assignment by specifying which problems they work on first. Some problems do not specify a method for solving. Students solve these problems using the RDW approach used for Application Problems. Lesson 11: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Demonstrate possible whole number side lengths of rectangles with areas of 24, 36, 48, or 72 square units using the associative property. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.28 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 3• 4 Student Debrief (10 minutes) Lesson Objective: Demonstrate possible whole number side lengths of rectangles with areas of 24, 36, 48, or 72 square units using the associative property. The Student Debrief is intended to invite reflection and active processing of the total lesson experience. Invite students to review their solutions for the Problem Set. They should check work by comparing answers with a partner before going over answers as a class. Look for misconceptions or misunderstandings that can be addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the lesson. You may choose to use any combination of the questions below to lead the discussion. Turn your paper horizontally and look at Problem 1. What property does this show? Share your answer to Problem 2. Discuss your answer to Problem 4 with a partner. What would the rectangle look like if the difference between side lengths was 0? How do you know? Compare your answer to Problem 5(c) with a partner’s. Did you both come up with the same side lengths? Why or why not? Explain to a partner how to use the strategy we learned today to find all possible side lengths for a rectangle with an area of 60 square units. Exit Ticket (3 minutes) After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help you assess the students’ understanding of the concepts that were presented in the lesson today and plan more effectively for future lessons. You may read the questions aloud to the students. Lesson 11: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Demonstrate possible whole number side lengths of rectangles with areas of 24, 36, 48, or 72 square units using the associative property. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.29 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 Problem Set 3 Name Date 1. The rectangles below have the same area. Move the ( ) to find the missing side lengths. Then solve. _____ cm 6 cm 1 cm b. 8 cm ______cm 2 cm a. Area: 8 × _____ = ______sq cm c. Area: 8 × 6 = (2 × 4) × 6 =2×4×6 = _____ × _____ 4 cm d. Area: 8 × 6 = (4 × 2) × 6 =4×2×6 = _____ × _____ = ______ sq cm _____ cm e. Area: 8 × 6 = 8 × (2 × 3) =8×2×3 = _____ × _____ = ______ sq cm ______ cm Area: 1 × 48 = ______ sq cm ______ cm = ______ sq cm 2. Does Problem 1 show all the possible whole number side lengths for a rectangle with an area of 48 square centimeters? How do you know? Lesson 11: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Demonstrate possible whole number side lengths of rectangles with areas of 24, 36, 48, or 72 square units using the associative property. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.30 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 Problem Set 3 3. In Problem 1, what happens to the shape of the rectangle as the difference between the side lengths gets smaller? 4. a. Find the area of the rectangle below. 8 cm 9 cm b. Julius says a 4 cm by 18 cm rectangle has the same area as the rectangle in Part (a). Place ( ) in the equation to find the related fact and solve. Is Julius correct? Why or why not? 4 × 18 = 4 × 2 × 9 =4×2×9 = _____ × _____ = _____ sq cm cm c. Use the expression 8 × 9 to find different side lengths for a rectangle that has the same area as the rectangle in Part (a). Show your equations using ( ). Then estimate to draw the rectangle and label the side lengths. Lesson 11: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Demonstrate possible whole number side lengths of rectangles with areas of 24, 36, 48, or 72 square units using the associative property. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.31 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 Exit Ticket 3 Name 1. Find the area of the rectangle. 8 cm 8 cm Date 2. The rectangle below has the same area as the rectangle in Problem 1. Move the ( ) to find the missing side lengths. Then solve. _______ cm ______ cm Area: 8 × 8 = (4 × 2) × 8 =4×2×8 = _____ × _____ = ______ sq cm Lesson 11: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Demonstrate possible whole number side lengths of rectangles with areas of 24, 36, 48, or 72 square units using the associative property. 9/30/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 4.C.32