• Middle School Mathematics Grade 6 Unit 2

    Subject: Mathematics
    Grade: 6
    Timeline: 32
    Unit 2 Title: Data Analysis & Inequalities (Select Data about Us and CMP2 Common Core Investigations 3 & 5, Math Shell)

    Unit Overview:
    This unit will provide the opportunity for students to explore statistics as a process of data investigation centering around four interrelated components, as stated by Graham (1987):
    1. Formulating and posing key questions from which to collect data about
    2. Determining a plan and collecting the necessary data
    3. Analyzing the data from organizing to selecting the appropriate representation and looking for patterns
    4. Interpreting the results to identify relationships and connect back to the original question posed
    After establishing a basic understanding of these processes, student will move from working with pre-generated data to generating their own to perform the steps listed above.  They will be able to pose questions worthy of investigation, formulate a plan, and execute it to represent distribution of data using line plots, bar graphs, histograms, number lines, box plots, and coordinate graphs/planes. Furthermore, they will be able to take the representations they have created and locate patterns and relationships regarding the distribution to expand their understanding of statistical variability through the generation of five point summaries, the calculation of mean, analysis of and comparison of more than one of these to each other.
    As the unit concludes, in accordance with new common core standards, students will also have a deeper understanding of rational numbers and be able to graph inequalities based within plausible situations.   They will grow from working with only quadrant I on a coordinate plane to all four. They will be able to identify specific characteristics of each and locate points when given details about it.  Overall, this unit affords them the opportunity to come to logical, well reasoned conclusions based within and supported by their data in whatever means they are now able to determine is appropriate.  

    Unit Objectives:
    At the end of this unit, students should have mastered the following skills and abilities:
    • Understand and use the process of data investigation: posing questions, collecting and analyzing data distribution, and making interpretations to answer questions.
    • Represent distribution of data using line plots, bar graphs, histograms, coordinate graphs, five number summaries, and across a coordinate plane.  
    • Compute the mean, median, mode, and range of the data
    • Make informal decisions about which graph(s) and which measures of center (mean, median, or mode) and range may be used to describe a distribution of data
    • Develop strategies for comparing distributions of data
    • Relate and compare the signs of numbers in ordered pairs to their locations in quadrants of coordinate planes and on number lines.
    • Distinguish comparisons of absolute value from statements about order
    • Write and graph inequalities to represent real world situations
    • Summarize numerical data sets by giving quantitative measures of center and variability.
    • Calculate a data set’s mean absolute deviation 

    Focus Standards:
    PA.CCSS.Math.Content.CC.2.4.6.B.1 Use a set of numerical data to develop an understanding of and recognize statistical variability. (6.SP.1) (6.SP.2) (6.SP.3)
    PA.CCSS.Math.Content.CC.2.4.6.B.2 Use numerical data and apply statistical properties to summarize and describe a distribution. (6.SP.4) (6.SP.5.a) (6.SP.5.b) (6.SP.5.c) (6.SP.5.d) (6.SP.5c)
    PA.CCSS.Math.Content.CC.2.1.6.E.2 Identify and choose appropriate processes to compute fluently with multi-digit numbers. (6.NS.2) 
    PA.CCSS.Math.Content.CC.2.1.6.E.4 Apply and extend previous understandings of numbers to the system of rational numbers. (6.NS.6a) (6.NS.6b) (6.NS.6c) (6.NS.7c) (6.NS.7d) (6.NS.8)
    PA.CCSS.Math.Content.CC.2.2.6.B.2 Understand the process of solving a one-variable equation or inequality and apply to real-world and mathematical problems. (6.EE.8)

    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.
    #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 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 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:
    • In a given real world mathematical situation, what data and/or situation is to best communicated on which type of graph. Additionally, how to justify the use of and analyze the results displayed on the selected method of representation.
    • What change (an increase or decrease) will occur in a median, mode, range, or mean based solely on the suggested addition of or subtract of provided numbers 
    • The purpose of and when it is appropriate to use a student calculated median vs. the mode vs. the mean vs. range of a data set when asked for mathematical reasoning to support a conclusion drawn.
    • How to set up a coordinate plane and plot points in all quadrants.
    • How to select the most useful scale/interval measure based on the data provided.
    • What the x and y axis represent on a graph and their purpose in plotting points
    • Absolute value of numbers (negative and positive)
    • When given a real world situation, how inequalities are written and their solution graphed on a number line.
    • The construction of a five-number summary and what the location of an individual number in that summary tells you about that data piece as compared to the data set.
    • How to graph an inequality and identify how many and what are possible solutions
    • The definitions of and which variable is independent and dependent
    • How and why measures of variability and “mean absolute deviation” can be used to summarize data.
    • Analyze numerical patterns 
    Competencies -Students will be able to:
    • Define and create a bar graph, line plot, coordinate graph, box plot, and histogram with an appropriate scale/interval(s) and labels based on the data set.
    • Calculate the modes of central tendency including mean, median, mode and range.
      • Predict how adding or subtracting numbers may affect the mean or median.
    • Plot and name points on a coordinate graph using the x and y coordinate in all quadrants.
      • Connect these points to create various polygons and calculate the length of the line segments between the points.
    • Correctly set up a coordinate graph’s scale and labels using a data distribution.
    • Identify the four quadrants and the characteristics specific to each one in order to correctly plot points.
    • Calculate and mathematically justify a predicted change in a mode of central tendency
    • Use zero at a center marker and create a number line for positive and negative numbers.
      • Correctly place rational numbers on the said number line
    • Find the absolute values of rational numbers
    • Write inequalities for given situations and graph the possible solution on a student generated number line.
    • Correctly identify the solution(s) to an inequality as an “open” or “closed” response i.e. is it part of the solution or not.
    • Identify the five-number summary of a student generated box plot
      • Define and locate the minimum value, lower quartile, median, upper quartile, and maximum value.
      • Locate individual piece of data in the summary 
      • Predict how the summary with change with changes to data.
    • Define measures of variability 
    • Calculate a data set’s mean absolute deviation after determining said data set’s mean. 

    • Written quiz after Data about Us Inv 1 & 2 plus Common Core Inv 3
    • Unit Exam 
    • Card Matching Sets from Math Shell
    • Daily informal assessment on opener as whole class
    •  “Do Now” mathematical review/introductory questions in the beginning of class daily
    • Optional “exit slips” at the end of class, lessons, etc.
    • 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, in accordance with the above, grade 5 does not go into any depth with this unit’s area of study.  The concepts within are introduced and developed during the sixth grade year to create a solid foundation from which students will draw upon in the coming academic years. 

    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)