• Middle School Mathematics Grade 6 Unit 4

    Subject: Mathematics
    Grade: 6 
    Timeline: 21 days
    Unit 4 Title: Area, Surface Area, and Volume (Select Covering & Surrounding and CMP2 Common Core Investigation 4)

    Unit Overview: 
    This unit will provide the opportunity for students to review key concepts from the end of their fifth grade year including reasoning about shape’s relationships to determine area, perimeter, and volume.  After recalling these formulas, they will begin work on finding the area of triangles (scalene, right, isosceles), special quadrilaterals, prisms and pyramids, at times, by drawing them to scale on a cm grid and by constructing nets of the given figures.  Furthermore, they will work to find the volume of right rectangular prisms with fractional edge lengths and representing the 3D figures using nets to assist in determining their surface area.  These requirements will afford them the opportunity to develop, model, discuss, and justify the formulas for finding the area of triangles, parallelograms, and 3D shapes.  Additionally, by the end of this unit, students will be able to recognize how triangles and parallelograms are mathematically connected and how elements such as the area of two objects and be the same which the perimeter of the same objects are different.  They  will gain the mathematical and problem solving abilities to determine which formula is required in a given real world situation, how to showcase this correctly labeling items such as base, height, side length, etc., and how to defend their logically drawn conclusions.

    Unit Objectives:
    At the end of this unit, students should have mastered the following skills and abilities:
    • If given constraints, determine if the area and perimeter of two figures will be the same.
    • Analyzing how the area of a triangle and the area of a parallelogram are related to the area of a rectangle
    • Developing formulas, strategies, and procedures, stated in words and/or symbols, for finding areas of rectangles, parallelograms, triangles and 3D shapes.  
    • Developing techniques for estimating the area and perimeter of an irregular figure.
    • Recognizing real-world situations in which measuring perimeter or area will help answer practical questions.
    • Representing 3D figures using nets made of rectangles and triangles using them to solve problems involving surface area. 

    Focus Standards:

    PA.CCSS. Math.Content.CC 2.3.6.A.1 – Apply appropriate tools to solve real-world and mathematical problems to involving area, surface area, and volume.  (6.G.1) (6.G.2) (6.G.3) (6.G.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 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 intoa 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. 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 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. Mathematically proficient students are also ableto compare the effectiveness of two plausible arguments, distinguish correct logic orreasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions.
    #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.   
    #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. In the elementary grades, students give carefully formulated explanations to each other
    #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 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:
    • Why perimeter, area, and volume have different labels of units (squared, cubed)
    • By what method(s) perimeter of triangles, rectangles, and parallelograms can be calculated
    • The relationship between rectangles and triangles as well as parallelograms and triangles in regards to determining their area.
    • How rotating a triangle changes the height and base.
    • When provided with a labeled example, what information is needed to calculate the area of triangle or parallelogram.
    • In what situations two non-congruent triangles or parallelograms can be created.  If two cannot be created, students will understand why the task is impossible and provide models to prove their conclusions.
    • Real world application of perimeter, (surface) area, and volume and when which is appropriate if given situations or models.
    • The role and importance of nets for representations and calculations involving 3D shapes.
    Competencies -Students will be able to:
    • Correctly label responses with units of measures as well as squared, cubed, etc.
    • Use rulers to find the side lengths of figures in order to apply the formula for perimeter
    • Utilize the formula(s) for area and volume when given the measurements and/or asked to find them.  
      • Sixth grade curriculum requires the inclusion of fractional side length of right rectangular prisms in determining volume.
    • Select which formula for area is appropriate based on the figure they are working with (i.e. rectangle vs triangle) and utilize it correctly.
    • Determine the area of a rectangle or triangle/ parallelogram or triangle if provided with one shape’s measurements and not the others.
    • Find the height and base of triangles and parallelograms.
    • Find the base or height or area of a triangle or parallelograms if provided with two of the three numbers.
    • Construct non-congruent triangles and parallelograms given numerous constraints, justify their drawings, and the rationale behind their findings.
    • Construct and/or use pre-generated nets to find the surface area of a shape 

    Assessments (Informal and Formal):
    • Unit Exam 
    • Daily informal assessment on opener as whole class
    • Optional exit slips to assess daily comprehension
    • 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.
    With specific regards to this unit, students were to use, describe and develop procedures to solve problems involving volume which was critical area 3.  They completed this process through a series of step by step lessons that allowed them to determine the formula for calculating volume when given both figures and/or measurements.  Furthermore, they were asked to explain why finding the volume was appropriate in these situations.  In order to best prepare them for sixth grade’s inclusion of fractional side lengths, they practiced finding volumes of right rectangular prisms with whole-number edge lengths in the context of solving real-world and mathematical problems. They also found volumes of solid figures composed of two non-overlapping right rectangular prisms.  

    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
    • Designing Triangles Under Constraints
    • Designing Parallelograms Under Constraints

    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)