• Middle School Mathematics Grade 6 Unit 5


    Subject: Mathematics
    Grade: 6
    Timeline: 19 days
    Unit 5 Title: Decimal and Percent Work (Investigation 1 – 3, 4.1, 4.2, and 5.1)

    Unit Overview: 
     
    This unit will provide the opportunity for students to expand upon and continue to the work begun in Bits and Pieces II.  The foundation has been laid and now students will take their knowledge of fractions and use this to make sense of decimals using many of the same principles they have become familiar with such as estimating to benchmarks, generalizing number patterns, and finding fraction/decimal equivalency based on place value interpretation.    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 equations (addition, subtraction, multiplication, or division) that involve decimals and evaluate the reasonableness of their responses.  In addition, students will gain the mathematical and problem solving abilities to determine which operation is required in a given real world situation.

    Unit Objectives:
     
    At the end of this unit, students should have mastered the following skills and abilities:
    • Using benchmarks and other strategies to estimate the reasonableness of results of operations with decimals.
    • Using estimates and exact solutions to make decisions
    • Looking for and generalize patterns in numbers
    • Using knowledge of decimals to develop algorithms for adding, subtracting, multiplying, and dividing.
    • Using the relationship between decimals and fractions to develop and understand why decimal algorithms work.
    • Recognizing place value interpretations to make sense of shortcut algorithms for operations.
    • Transferring their knowledge to real world calculations of measurement involving decimals.
    • Recognizing when addition, subtraction, multiplication, or division is the appropriate operation to solve a problem.
    • Solving problems using arithmetic operations on decimals.
    • Solving percentage problems of the form a% of b equals c for any one of the variables a, b,or c.
    Investigation 1:
    • Students will develop addition and subtraction of decimals.
    Investigation 2: 
    • Students will focus on developing an algorithm for multiplying decimals. 
    • Students will use fractions to help make sense of multiplication of decimals. 
    • Students will look at products, find missing factors
    • Students will use estimation as a way to determine where the decimal has to be in a product of decimal numbers
    Investigation 3:
    • Students will develop an algorithm for division of decimals.
    • In developing the algorithm, students will solve a set of contextualized problems that provide a common sense way to think about decimal division based on what they already know about whole-number and fraction division.
    • Students will use the fraction form of decimals to develop an algorithm for dividing decimals
    Investigation 4 (lessons 1 and 2):
    • Students will look at real (typical) situations in which one encounters percents. Typical situations of taxes  and tips help students think about taking a percent of a number and to consider the amount left when a reduction is made and the total when taxes are added.
    Investigation 5 (lesson 1): 
    • Students are asked to devise a general strategy for finding a percent when they are dealing with totals that are more than or less than 100.

    Focus Standards:
     
    PA.CCSS.Math.Content. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation (6.NS.3)
    PA.CCSS.Math.Content. Write, read, and evaluate expressions in which letters stand for numbers. (6.EE.2)
    PA.CCSS.Math.Content. Write expressions that record operations with numbers and with letters standing for numbers. (6.EE.2.a)
    PA.CCSS.Math.Content. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?  Use substitution to determine whether a given number in a specified set makes an equation or inequality true. (6.EE.5)
    PA.CCSS.Math.Content. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set (6.EE.6)
    PA.CCSS.Math.Content. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. (6.RP.3.c)

    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 analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. 
     
    #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. 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, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. 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.
     
    #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.
     
    #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. 

    Concepts - Students will know:
    • Estimating decimals/fractions using 0, 1/2, 1, 11/2, 2 as benchmarks.
    • Algorithms to add, subtract, multiply, or divide any computations involving decimals 
    • The role of place value in calculations and fraction equivalences with decimals
    • Decimal and fraction notation
    • Relationships between factors and products in decimal multiplication
    • Decimal placement when multiplying decimals using powers of 10
    • Repeating vs. terminating decimals
    • Find the total cost of an item(s) when given percentage of sales tax
    • Real world situations that involve percentages such as leaving tips or amount of savings on a sale or utilizing a coupon.
    • Determine the amount I am saving when something is on sale by using the given percentage of discount
    Competencies -Students will be able to:
    • Find the nearest benchmark of a given decimal(s) and estimate the sum or difference.
    • Solve operations (addition, subtraction, multiplication, division) that involve any combination of decimals, repeating or terminating.
    • Find the missing factor when given the product and one factor.
    • Place the decimal point in the appropriate place when calculating with decimals.
    • Define and write terminating decimals.
    • Define and write repeating decimals.
    • Determine what operation is appropriate in a given decimal situation
    • Write given decimals as fractions and vice versa.
    • Name the place values of numbers and determine the place value of decimals.
    • Name a decimal as a fraction according to its place value
    • Calculate the amount of a tip by using 5%, 10%, 15% or 20%.
    • Find the amount of tax on a purchase amount when the given the percentage of tax.
    • Find the percentage off on a purchase item(s) when given the percentage of the discount.
    • Find the percent saved when using coupons that give a specific dollar amount
    • Solve a percentage problem by working backwards 

    Assessments:
    • 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.
    • Unit Exam 
    • “Do Now” mathematical review/introductory questions in the beginning of class daily
    • Optional “exit slips” at the end of class, lessons, etc.

    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.
     
    The writers of this series have offered 6th grade teachers numerous ways to contend with the challenges students experience with fractional and decimal work.  Prior to beginning work in this area, the book highlights the mathematical goals and expectations for the students in order to master these topics.  They will serve as (a) continuous guidelines and reference points for both teacher and student to refer to, frequently, in the form of learning targets and opportunities to informally ask students to reflect on what they are doing and in what matter.  Additionally, they are connected to the standards aligned to the 5th grade series.  Exemplary teaching will constantly seek out ways for students to connect these to their previous mathematical skills and provide a variety of instructional practices that will allow students to make sense of and make generalizations regarding the relationships between fractions, decimals, and percents.  
     
    The actual investigations themselves are progressive and build upon the previous one to allow a gradually increase in abilities based on repeated procedural skill, comprehension, and reasoning, lesson by lesson, within each investigation. This component alone is significantly important in this book as students can struggle with these topics.    Each “Getting Ready” (included in both the teacher and student guide) draws on the knowledge gained in Bits and Pieces II and previous lessons.  Furthermore, they lend themselves to whole class example or partner work into a more independent study type structure to which an instructor should bring the class back together at the end of session and assess student understanding through discussion, student examples/work, etc.  
     
    The ACE (Applications—Connections—Extensions) practice problems offer three levels of out of the classroom work centered on the topics, goals, and targets of each investigation.  Based upon some of the informal assessments stated above and teacher observations, the ACE problems can be tailored to each group of learners and their needs.  The options for homework problems in the form of ACE problems are as follow:
     
    a) Applications: These exercises help students solidify their understanding by providing practice with ideas and strategies that were in the 
    Investigation.  Applications contain contexts both similar to and different from those in the Investigation.
    b) Connections: A powerful learning strategy is to connect new knowledge to prior learning. The Connections section of the homework provides this opportunity. This section also provides continued review of concepts and skills across the grades.
    c) Extensions: These exercises may provide a challenge for students to think beyond what is covered in the problems in class. 

    Differentiation:
     
    Each lesson has differentiation options for each portion of the lesson. Additional differentiation options are listed with directions and student masters in the Teacher’s Guide.
    • 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 
    • Learning Target
    • Students will have an opportunity to  Explore the lesson (independently, with a partner , small group)

    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)