• Elementary Mathematics Grade 5 Unit 4

    Subject: Mathematics
    Grade: 5
    Timeline: 19 days
    Unit 4 Title: Division of Multi-Digit and Decimal Numbers

    Unit Overview: 
    This unit will give the opportunity for students to practice, review, and develop strategies for division of whole numbers and decimals. The students will be engaged in activities with division facts, extensions, patterns, divisibility and remainders, with a major focus on the partial quotients standard algorithm.  The unit will also provide review and practice for multiplication methods of whole numbers and decimals.

    Unit Objectives:
    At the end of this unit, all students must be able to demonstrate knowledge of division of whole numbers and decimals to the hundredths.  Students must be able to divide up to four-digit dividends and two-digit divisors.  Students will use different methods to illustrate and explain their calculations. Students must understand the relationship between multiplication and division.

    Focus Standards:
    PA.CCSS.Math.Content.CC.2.1.5.B.1 Apply place value concepts to show an understanding of operations and rounding as they pertain to whole numbers and decimals. (5.NBT.1)
    PA.CCSS.Math.Content.CC.2.1.5.B.2 Extend an understanding of operations with whole numbers to perform operations including decimals. (5.NBT.6, 5.NBT.7)
    PA.CCSS.Math.Content.CC.2.4.5.A.1 Solve problems using conversions within a given measurement system.. (5.MD.1)

    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 jumping into a solution attempt. They monitor and evaluate their pathway and change course if necessary. Mathematically proficient students check their answers to problems using a different method and continually ask themselves, “Does this make sense?” 
    #2 Reason Abstractly and Quantitatively.
    Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complimentary abilities to bear on problems involving quantitative relationships: the ability to decontextualize and the ability to contextualize. Quantitative reasoning entails habits of creating coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 
    #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 make conjectures and build a logical progression of statements to explore the truth of their conjectures. 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. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. 
    #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 early grades, this might be as simple as writing an addition equation to describe a situation.  
    #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 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 use clear definitions in discussion with others and in their own reasoning. They are careful about specifying units of measure, specifying axes and use careful calculations. 
    #7 Look for and make use of structure.  
    Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. 
    #8 Look for and express regularity and 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. 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:
    • The definitions of dividend, divisor and quotient 
    • Partial Quotient Algorithm
    • Function of a remainder
    • Basic division facts and extensions
    Competencies -Students will be able to:
    • Divide whole numbers and decimals using four-digit divisors and two-digit dividends
    • Become fluent with basic division facts and extensions
    • Demonstrate the relationship between multiplication and division
    • Work with and interpret remainders
    • Solve word problems that include division

    • Unit 4 Assessment
    • Daily RSA

    Elements of Instruction:
    With the Grade 4 Common Core State Standards for Operations and Algebraic Thinking, 4.OA.3, students must be able to solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted.  According to the Common Core State Standards for, Operations and Algebraic Thinking, 4.OA.4, students must be able to find all factor pairs for a whole number in the range of 1-100. This foundation is essential for this unit of study.

    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 to Games. These games are foundational for this unit and should be used with students who are struggling with these foundations.
    • Multiplication Top-It (Accurate Multiplication Computation)
    • Multiplication Wrestling (Accurate Multiplication Computation)
    • Broken Calculator  (Understanding Meanings of Operations)
    • Subtraction Target Practice  (Making Reasonable Estimates)

    Interdisciplinary Connections:
    Grade 5 utilizes the American Tour to connect mathematics across the content areas including Social Studies, Geography and English Language Arts. Students identify data organized in text, tables, graphs and maps. Students read and answer questions about the U.S. Census. They use map scales to estimate real distances along straight paths. Students interpret information in the text and solve problems about the changing emphasis on mathematics instruction. They explore and practice concepts using fractions, decimals and percents. Students use contour maps of precipitation and growing seasons to compare the climates for various locations in the U.S. they compare data in the American Tour by constructing and analyzing a line graph. They examine various representations of the total population and area of the U.S. Students use percents to estimate the number of rural and urban Americans for various years. Finally, they read information about Yellowstone National Park and Old Faithful, solve problems and generate data for a graph based on a formula for predicting when Old Faithful will erupt. 

    Additional Resources / Games:
    Students will play a variety of games that directly support the content of the lesson and the overall goals for the unit. Games for unit four include:
    • Division Top-It
    • Division Arrays
    • Factor Captor
    • Divisibility Dash