• Elementary Mathematics Grade 4 Unit 3

    Subject: Mathematics
    Grade: 4 
    Timeline: 9 days
    Unit 3 Title: Division

    Unit Overview: 
    This unit will give students the opportunity to further develop their understanding of multiplication operations, while using that knowledge to develop strategies for dividing whole numbers, solving division number stories, and solving extended multiplication and division facts.  The algorithm that students will learn to divide is "partial-quotients."  We encourage students to use this algorithm only, because it reinforces place value concepts, and all class work will use this algorithm.  Students will use their developing proficiency with division to solve number stories.  Students will also further develop their understanding of number patterns.  Students who are struggling with their multiplication/division facts should be practicing each night at home.

    Unit Objectives:
    At the end of this unit, students must be able to divide multi-digit numbers and interpret remainders in division number stories.   

    Focus Standards:
    PA.CCSS.Math.Content.CC.2.1.4.B.2 Use place value understanding and properties of operations to perform multi-digit
     arithmetic. (4.NBT.6)
    PA.CCSS.Math.Content.CC.2.2.4.A.1 Represent and solve problems involving the four operations.(4.OA.2, 4.OA.3)
    PA.CCSS.Math.Content.CC.2.2.4.A.2 Develop and/or apply number theory concepts to find factors and multiples.(4.OA.4)
    PA.CCSS.Math.Content.CC.2.2.4.A.4 Generate and analyze patterns using one rule. (4.OA.5)

    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 monitor and evaluate their progress and change course if necessary. Mathematically proficient students check their answers to problems using a different method, and they 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 complementary abilities to bear on problems involving quantitative relationships:  the ability to decontextualize-to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents-and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.  Quantitative reasoning entails habits of creating a 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 a 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. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.
    #5 Use appropriate tools strategically.  
    Mathematically proficient students in First Grade have access to a variety of concrete (e.g. 3-dimensional solids, ten frames, number balances, number lines) and technological tools (e.g., virtual manipulatives, calculators, interactive websites) and use them to investigate mathematical concepts. They select tools that help them solve and/or illustrate solutions to a problem. They recognize that multiple tools can be used for the same problem- depending on the strategy used. For example, a child who is in the counting stage may choose connecting cubes to solve a problem. While, a student who understands parts of number, may solve the same problem using ten-frames to decompose numbers rather than using individual connecting cubes. As the teacher provides numerous opportunities for students to use educational materials, first grade students’ conceptual understanding and higher-order thinking skills are developed.
    #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.  Later, students will see 7 * 8 equals   7 * 5 + 7 * 3, in preparation for learning about the distributive property.

    Concepts - Students will know:
    • division vocabulary
    • the partial quotients algorithm
    • how to interpret remainders in number stories
    Competencies -Students will be able to:
    • define all parts of a division problem
    • divide up to 4-digits by 1-digit
    • choose what to do with a remainder when given a number story

    • Unit 3 Progress Check
    • Daily RSA
    • Division Fact Tests
    • Division Quizzes

    Elements of Instruction:
    Learners in 4th grade will extend their understanding of multiplication to the inverse operation, division.  Students will need to understand the algorithm that is presented throughout this unit, and be successful in solving multi-digit division problems. 

    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.  There is also an option for periodic 50-fact division practice tests

    Interdisciplinary Connections: 
    • Mental Math and Math message routines

    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 three include:
    • Division Arrays
    • Division Dash
    • Multiplication Wrestling
    • Multiplication Top-It
    A constant practice with basic multiplication can be given daily. Example: 50 Facts Test