• Middle School Mathematics Grade 6 Unit 1

    Subject: Mathematics
    Grade: 6 
    Timeline: 26 days
    Unit 1 Title: Factors, Multiples, and Fractional Work (Select Prime Time and Bits & Pieces II)

    Unit Overview: 
    This unit will provide the opportunity for students to build upon key concepts from the end of their fifth grade year including work with adding and subtracting fractions.  This unit begins with work finding factors and multiples in a variety of real world situations.  Relying on their previous knowledge, they will continue their begun work on multiplying mixed numbers and (improper) fractions.  The model “Keep, change, flip” has proved effective for dividing and is rooted in a strong working knowledge of converting mixed numbers to improper fractions and using multiplication.  Therefore, this is briefly reviewed as a prerequisite skill necessary to achieve success with this area of the “major” clusters in the scope of 6th grade focus content.  They will rely upon methods of their own developing using several models (brownie pan, number line, etc) as tools.  By the end of this unit, they will be able to design and utilize efficient  and precise algorithms that make sense in their computational understanding and allow them to solve any multiplication or division equations that involve factions (like and unlike denominations), improper fractions, mixed numbers and whole numbers.  In addition, they will be able to determine common multiples and factors of number sets as well as the set’s GCF and LCM in numbers up to 100.  Students will gain the mathematical and problem solving abilities to determine which operation is required in a given real world situation and whether an estimation, based in part on benchmarks, will be enough or if an exact answer is required based on the information provided.  

    Unit Objectives:
    At the end of this unit, students should have mastered the following skills and abilities:
    • Developing understanding of factors and multiples, common factors and common multiples, and the relationship among them.
    • Determining the LCM and GCF of number sets up to 100 in real world situations.
    • Using benchmarks and other strategies to estimate the reasonableness of results of operations with fractions.
    • Developing ways to model quotients with areas, strips, and number lines based off possible situations they may encounter in their lives.
    • Using estimates and exact solutions to make decisions
    • Looking for and generalize patterns in numbers
    • Using knowledge of fractions and equivalence of fractions to develop algorithms for multiplying, and dividing fractions.
    • Recognizing when multiplication or division is the appropriate operation to solve a problem.
    • Writing fact families to show the inverse relationship between multiplication and division.
    • Solving problems using arithmetic operations on fractions.

    Focus Standards:
    PA.CCSS.Math.Content.2.1.6.E.2 - Apply and extend previous understandings of multiplication and division to divide fractions by fractions. (6.NS.2) (6.NS.3) 
    PA.CCSS.Math.Content.2.1.6.E.3 - Develop and/or apply number theory concepts to find common factors and multiples. (6.NS.4)

    Mathematical Practice Standards:
    #1 Make sense of problems and persevere in solving them. 
    Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. They check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
    #3 Construct viable arguments and critique the reasoning of others. 
    Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
    #4 Model with mathematics. 
    Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
    #5 Use appropriate tools strategically. 
    Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, or a calculator. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
    #6 Attend to precision. 
    Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.
    #7 Look for and make use of structure. 
    Mathematically proficient students look closely to discern a pattern or structure. Mid-level students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property.
    #8 Look for and express regularity in repeated reasoning. 
    Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. 

    Concepts - Students will know:
    • How finding the LCM and GCF be necessary and helpful in real world situations.
    • Fractional “fact families” 
    • Algorithms for multiplication or division of any computations involving fractions with like or unlike denominators
    • Operation selection in given real world situations.
    • Varies strategies to determine the LCM and GCF
    • When two responses (fractions) are equivalent i.e. if one is the reduced version of the other.
    Competencies -Students will be able to:
    • Find the LCM or GCF for sets of numbers.
    • Solve operations (multiplication, division) that involve any combination of fractions with like and unlike denominations, mixed numbers, whole numbers, and improper fraction utilizing strategies such as the brownie pan model, finding equivalent fractions, and number lines.
    • Complete fact families by finding missing number sentences and/or values.
    • Convert mixed numbers into improper fractions before finding the product
    • Write number sentences that reflect a given problem.
    • Write algorithms to multiply or divide
    • Simplify fractions into their lowest form 

    • “Do Now” mathematical review/introductory questions in the beginning of class daily
    • Optional “exit slips” at the end of class, lessons, etc.
    • Unit Exam 
    • Daily informal assessment on opener as whole class
    • Collection/Grading of homework problems to monitor ongoing progress.
    • Reflection questions at the end of each investigation to assess students’ comprehension of key concepts.

    Elements of Instruction:
    As stated on the common core state standards initiative website (http://www.corestandards.org/Math/Content/5/introduction), Grade 5, instructional time should focus on three critical areas: (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and (3) developing understanding of volume.
    Therefore, fifth grade lays the groundwork for 6th grade concepts regarding fractional operations as our task is to “extend” the mathematical topics covered the previous year.  Although the skill set necessary to solve addition and subtraction problems with fractions are not all the same, the introduction to reducing, comparing, as well as those two operations and the beginnings of multiplication and division, are vital to the successful mastery during their 6th grade course work.  CMP2 seeks to provide students with several strategies to solve division of a fraction by a fraction which requires converting a mixed number to an improper fraction and multiplying.  Each step for this process is rooted in knowledge gained their 5th grade year.  Although, as stated above, students will have been exposed to fractional work prior to this academic year, there could be several issues that occur based off the same two mathematical principles and practices.
    1. Lacking a basic knowledge of multiplication facts which can cause any of the following including an inability to convert mixed numbers into improper fractions, the inability to use the common practice of dividing fractions using the “KCF” strategy, and  the inability to list common multiples of two or more numbers in order to find the LCM.
    2. Lacking a basic knowledge of divisibility rules that prevent students from simplifying fractions into the lowest form, prevent students from listing all possible factors of a number in order to find the GCF, and, most obviously, prevent the actual division of a fraction by a fraction.
    Covering, in both instruction and student practice work, all the necessary and suggested prerequisite skills and activities will serve to remind students of where they left off at.  Additionally, it will provide the instructor with an excellent gauge as to where their students’ strengths and weaknesses are.  Knowing your students levels and thought processes throughout each lesson during discussion, cooperative learning, and independent work will allow the teacher to prevent or correct these as or when they arise.

    Each lesson and/or unit offers a wide variety of ways to differentiation for all levels of learners.  These include:
    • Special Need Handbook (adapting instruction/lessons)
    • Unit Projects
    • Spanish Additional Practice and Skills guide
    • Strategies for English Language Learners Guide
    • “Extension” homework questions

    Interdisciplinary Connections:
    • Mathematical Reflections
    • “Did You Know?” sections
    • phschool.com and web codes
    • “Connections” homework questions
    • The real-world context embedded in lesson problems 

    Additional Resources / Games:
    CMP2 and/or the Erie School District provide the following additional resources to aid students in achieving mathematical success. 
    • Additional Practice worksheets per investigation
    • Skills Review worksheets to target key components of each investigation
    • Parent letter to be sent home prior to beginning the unit to share with parents the skills, goals, and 
    • expectations of the coming unit.
    • Assessment Resources workbook with extra test items (multiple choice, essay, open ended, 
    • question bank, etc)
    • Investigation specific pre-generated notebooks that include tables, graphs, problem numbers, and all 
    • other items students may need to complete the investigation and all its parts.  Students are provided
    • with one per unit. 
    • Reflection questions at the end of each investigation to assess students’ comprehension of key
    • concepts. 
    • phschool.com and web codes
    • Transparencies of models, graphs, etc used within lesson(s)