• # Middle School Mathematics Grade 8 Unit 3

Subject: Mathematics
Timeline: 30 days
Unit 3 Title: Kaleidoscopes, Hubcaps, and Mirrors

Unit Overview:

Kaleidoscopes, Hubcaps, and Mirrors, the last geometry and measurement unit in the Connected Mathematics 2 Curriculum, help students to refine their knowledge of symmetry and use it to make mathematical arguments.  The purpose of this unit is to stimulate and sharpen students’ awareness of symmetry, congruence, their connections, and to begin to develop their understanding of the underlying mathematics.

Unit Objectives:

At the end of this unit, all students must:
• Understand important properties of symmetry.
• Recognize and describe symmetries of figures.
• Use tools to examine symmetries and transformations.
• Make figures with specified symmetries.
• Identify basic design elements that can be used to replicate a given design.
• Perform symmetry transformations of figures, including reflections, translations, and rotations.
• Examine and describe the symmetries of a design made from a figure and its image(s) under a symmetry transformation.
• Give precise mathematical directions for performing reflections, rotations, and translations.
• Draw conclusions about a figure, such as measures of sides and angles, lengths of diagonals, or intersection points of diagonals, based on symmetries of the figure.
• Understand that figures with the same shape and size are congruent.
• Use symmetry transformations to explore whether two figures are congruent.
• Give examples of minimum sets of measures of angles and sides that will guarantee that two triangles are congruent.
• Use congruence of triangles to explore congruence of two quadrilaterals.
• Use symmetry and congruence to deduce properties of figures.
• Write coordinate rules for specifying the image of a general point (x, y) under particular transformations.

Focus Standards:

PA.CCSS.Math.Content.CC.2.3.8.A.1. Understand and apply congruence and similarity using various tools. (8.G.A.1, 8.G.A.2, 8.G.A.3, 8.G.A.4, 8.G.A.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 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.

#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.

#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 × 15 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:
• Important properties of symmetry: reflection, rotation, and translation
• Appropriate use of tools for examining symmetries and transformations
• How to make figures with pre-determined symmetries
• Directions for performing reflections, rotations, and translations
• How to draw conclusions about a figure, such as measures of sides and angles, lengths of diagonals, or intersection points of diagonals based on symmetries of the figure.
Competencies -Students will be able to:
• Utilize important properties of symmetry
• Recognize and describe symmetries of figures
• Use tools to examine symmetries and
• transformations
• Make figures with specified symmetries
• Identify basic design elements that can be used to replicate a given design
• Perform symmetry transformations of figures, including reflections, translations, and rotations
• Examine and describe the symmetries of a design made from a figure and its image(s) under a symmetry transformation
• Give precise mathematical directions for
• performing reflections, rotations, and
• translations
• Draw conclusions about a figure, such as measures of sides and angles, lengths of diagonals, or intersection points of diagonals, based on symmetries of the figure.

Assessments:

Formative Assessments:
• Informal assessments on learning targets and additional practice pages
• Check-up Quiz 1
• Partner Quiz
• Check-Up Quiz 2
Summative Assessment:
• Common Core Unit Assessment - Kaleidoscopes, Hubcaps, and Mirrors

Elements of Instruction:

As mentioned previously, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.

Representing and reasoning about patterns of change has been the main focus up to this point in the development of algebra.  Students have used tables, graphs, and symbols to represent relationships and to solve equations to find information or make predictions.  The properties of real numbers such as the Commutative and Distributive properties were first introduced in Accentuate the Negative and then used again in Moving Straight Ahead and Growing, Growing, Growing.  The Distributive Property was expanded in Frogs, Fleas, and Painted Cubes to include multiplication of two binomials.

Major misconceptions by and struggles for students in this unit include:
1. Adding  rate of change to y-intercept in the equation
2. Trying to solve exponential equation by adding or subtracting, or by using square root
3. Solving for the base of exponents by dividing
4. Aligning the y-scale on a graph to the numbers on the table; therefore creating a straight line
5. Finding the y-intercept in a table

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 Needs Handbook
• Unit Projects
• Spanish Additional Practice and Skills guide
• Strategies for English Language Learners Guide
• “Extension” homework questions
• District-created notebooks

Interdisciplinary Connections:
• Mathematical Reflections
• “Did You Know?” sections
• phschool.com and web codes
• “Connections” homework questions
• The real-world context embedded in lesson problems