Unit 2 Algebraic Expressions Applied to Geometry

Transcription

1 Find Unit 2 Algebraic Expressions Applied to Geometry Number of Days: 24 10/17/16 11/18/16 Unit Goals Stage 1 Unit Description: Students will apply expressions to two-dimensional and three-dimensional geometry. Students will evaluate and translate expressions and use properties of arithmetic (commutative, associative, and distributive) to create equivalent expressions. Students will apply expressions to two-dimensional shapes by exploring and finding the area of triangles and quadrilaterals. Students will also solve for unknowns in three-dimensional shapes by finding surface area and volume of right rectangular prisms with fractional edge lengths. Materials: algebra tiles*, unit cubes, centimeter cubes, calculators (* Paper tool available at LBUSD Instructional Tools Middle School Mathematical Tools) Standards for Mathematical Practice SMP 1 SMP 2 SMP 3 SMP 4 SMP 5 SMP 6 SMP 7 SMP 8 Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Standards for Mathematical Content Clusters Addressed [m] 6.EE.A Apply and extend previous understandings of arithmetic to expressions. [s] 6.G.A Solve real world and mathematical problems involving area, surface area, Transfer Goals Students will be able to independently use their learning to Make sense of never-before-seen problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Making Meaning UNDERSTANDINGS Students will understand that Algebraic expressions can be used to represent real life situations. Expressions are equivalent when they name the same number regardless of which number the variable stands for. The area of polygons can be found by decomposing the polygon into triangles and rectangles. A three-dimensional figure can be represented using a net to visualize its faces and dimensions. Using the formulas for surface area and volume can help solve realworld problems. The volume of a right rectangular prism with fractional side lengths can be determined by packing or filling the solid with unit cubes. This yields the same results as using the formula for finding volume. KNOWLEDGE Students will know The definition of academic vocabulary words, such as variables, expressions, composite figure, surface area, nets, and fractional edge lengths. Variables represent a quantity that can change. ESSENTIAL QUESTIONS Students will keep considering How can expressions be used to represent and solve mathematical and real life situations? How can decomposing area formulas help find the area? How does understanding the volume of a right rectangular prism with whole number side lengths connect to the volume of a right rectangular prism with fractional side lengths? Acquisition SKILLS Students will be skilled at and/or be able to Use mathematical vocabulary to describe expressions. Evaluate expressions and formulas in real-world problems. Create equivalent expressions by applying properties of operations. Identify when expressions are equivalent using properties of operations. Use the formulas for area, surface area, and volumes of triangles, quadrilaterals, polygons, cubes, and right rectangular prisms to solve real-world and mathematical problems. the volume of a right rectangular prism with fractional side lengths by and volume. packing or filling the solid with unit cubes and by multiplying the edge lengths. LONG BEACH UNIFIED SCHOOL DISTRICT 1 Reposted 11/16/16

2 Assessed Grade Level Standards Standards for Mathematical Practice SMP 1 Make sense of problems and persevere in solving them. SMP 2 Reason abstractly and quantitatively. SMP 3 Construct viable arguments and critique the reasoning of others. SMP 4 Model with mathematics. SMP 5 Use appropriate tools strategically. SMP 6 Attend to precision. SMP 7 Look for and make use of structure. SMP 8 Look for and express regularity in repeated reasoning. Standards for Mathematical Content [m] 6.EE.A Apply and extend previous understandings of arithmetic to expressions. 6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers. a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation Subtract y from 5 as 5 y. b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms. c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real world problems. Perform arithmetic operations, including those involving whole number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s³ and A = 6 s² to find the volume and surface area of a cube with sides of length s = 1/2. 6.EE.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. 6.EE.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. [s] 6.G.A Solve real world and mathematical problems involving area, surface area, and volume. 6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. 6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems. 6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. Key: [m] = major clusters; [s] = supporting clusters; [a] = additional clusters LONG BEACH UNIFIED SCHOOL DISTRICT 2 Reposted 11/16/16

3 Assessment Evidence Unit Assessment Evidence of Learning Stage 2 Claim 1: Students can explain and apply mathematical concepts and carry out mathematical procedures with precision and fluency. Concepts and skills that may be assessed in Claim1: 6.EE.A The student uses mathematical terms to describe expressions. The student evaluates expressions and expressions from formulas in real-world problems. The student creates equivalent expressions by applying properties of operations. The student identifies when expressions are equivalent by utilizing properties of operations. 6.G.A The student determines the area of triangles, special quadrilaterals, and polygons using composition and decomposition in solving real-world and mathematical problems. The student determines the volume of right rectangular prisms with fractional edge lengths in solving real-world and mathematical problems. The student determines the surface area of three-dimensional figures formed by nets of polygons in the context of solving real-world and mathematical problems. Claim 2: Students can solve a range of wellposed problems in pure and applied mathematics, making productive use of knowledge and problem-solving strategies. Standard clusters that may be assessed in Claim 2: 6.EE.A G.G.A Other Evidence Formative Assessment Opportunities Claim 3: The student can clearly and precisely construct viable arguments to support their own reasoning and critique the reasoning of others. Standard clusters that may be assessed in Claim 3: 6.EE.A Claim 4: The student can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Standard clusters that may be assessed in Claim 4: 6.G.A s Informal teacher observations Checking for understanding using active participation strategies Exit slips/summaries Modeling Lessons (SMP 4) Formative Assessment Lessons (FAL) Quizzes/Chapter Tests Intranet Unit Assessment Resources Big Ideas Math Performance s SBAC Interim Assessment Blocks Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website M Mathematics Curriculum Documents LONG BEACH UNIFIED SCHOOL DISTRICT 3 Reposted 11/16/16

4 I will explore expressions by participating in the Opening. OPENING TASK Secret Number Expressions This Opening is a riddle/puzzle. After having the students do the riddle/puzzle in groups, have them discuss if they see any patterns. What is constant? What is changing? What is happening? Have them record their findings. Share as a whole group. This activity is a gateway into the entire unit. (SMP 7 and 8) Secret Number Expressions I will use variables to represent numbers by Explaining that a variable can represent an unknown number. (SMP 3) o What does a variable represent? o Reflect on the opening task. How does it relate to variables? Variable Activity 3-4 I will read, write, and evaluate expressions by Identifying parts of an expression, such as coefficient, constant, term, variable, and operation words. Translating expressions from symbolic to verbal and verbal to symbolic. (SMP 2) Representing expressions with algebra tiles. (SMP 5) Evaluating expressions at specific values for their variables. o How do you read and write expressions? o What does it mean to evaluate an expression? Section 3.1 (Examples 1, 2, 3, 5, and 6) Section 3.2 (Activities 2 and 3; Examples 1, 2, 3, and 4) STEM Video: Shadow Drawings Reference Tool for Algebra Tiles Which One Doesn t Belong? Algebraic Expressions Dominoes: Writing Algebraic Expressions Mix-Freeze-Match: Expressions Rectangle Perimeter 1 STEM Performance : Describing Change LONG BEACH UNIFIED SCHOOL DISTRICT 4 Reposted 11/16/16

5 I will generate equivalent expressions by 3-4 Using properties of operations, such as commutative, associative and distributive. Combining like terms with nonnegative coefficients. Using algebra tiles. (SMP 5) Explaining that equivalent expressions name the same number regardless of which number the variable stands for. (SMP 3) o Provide a real-world situation which can be represented with numerical or expressions that can be rewritten using one of the mathematical properties. (SMP 2) o How can you use algebra tiles to represent combining like terms? Section 3.3 (Activities 2 and 3; Examples 1, 2, and 3) Section 3.4 (Activity 1; Examples 1, 2, 4, and 5) Extension 3.4 (Examples 1, 2, and 3) Algebraic Expressions Equivalent Expressions Equivalent Expressions 1 Equivalent Expressions 2 Product of Distributive Property Distance to School Lacrosse Performance Perimeter of Geometric Figures Performance Rectangle Perimeter I will apply my knowledge of expressions to explore the area of polygons by Discovering the formulas for triangles and special quadrilaterals by composing into rectangles. (SMP 7) Applying the discovered formulas of area of parallelograms, triangles, and trapezoids to solve real-life and mathematical problems. Decomposing polygons into triangles and quadrilaterals, using known area formulas, and adding them to find the total area. (SMP 7) o When is area used in real life? o Why is the formula for the area of a parallelogram the same as the formula for the area of a rectangle? o Why are you not able to use one simple formula to find the area of a composite figure? o Can you decompose a composite figure only one way? Justify your reasoning. Section 4.1 (Activities 1 and 2; Examples 1, 2, and 3) Section 4.2 (Activities 1 and 2; Examples 1, 2, and 3) Section 4.3 (Activity 1; Examples 1 and 3) Extension 4.3 (Examples 1 and 2) STEM Video: Gold Course Maintenance Deriving the Area Formulas Finding Areas of Polygons Desmos: Exploring Triangle Area with Geoboards Game Closet: Pick Your Polygon Game Closet: Math Card War Areas of Special Quadrilaterals Concert Stages Performance Wallpaper Decomposition LONG BEACH UNIFIED SCHOOL DISTRICT 5 Reposted 11/16/16

6 I will check my understanding of expressions by participating in the FAL. I will apply my knowledge of expressions to explore surface area of threedimensional figures by OPTION #1: FORMATIVE ASSESSMENT LESSON Representing the Laws of Arithmetic (SMP 1, 2, 3, 5, 6, 7, 8) Representing with nets made up of rectangles and triangles. Solving real-life and mathematical problems. (SMP 2) Using the nets to find the individual areas of the faces and adding them to find the total surface area. o How can you use nets to find surface area? o Describe a scenario where you would need to find the surface area of a three-dimensional figure. Section 8.2 (Activities 1, 2, and 3; Examples 1 and 2) Section 8.3 (Activities 1, 2, 3,and 4; Examples 1 and 2) STEM Video: Packaging Design Nets Template Parallelogram Tents Performance Modeling: Dandy Candies 3-4 I will apply my knowledge of expressions to explore volume of right rectangular prisms with fractional edge lengths by Using unit cubes to fill or pack the solid. Explaining that filling or packing the solid with unit cubes yields the same result as multiplying the edge lengths of the prism. (SMP 3) Applying either formula, V = Bh and V = lwh. Solving real-life and mathematical problems. (SMP 2) o How can you apply your knowledge of volume of right rectangular prisms with fractional edge lengths to those with decimal edge lengths? Explain. (SMP 3) o Compare and contrast surface area and volume. Section 8.4 (Examples 1 and 2) Building Conceptual Understanding of Volume with Fractional Edge Lengths Powerpoint Sugar Cubes Volumes with Fractional Edge Lengths Building Blocks Volume of Rectangular Prisms Volume of Rectangular Prisms with Fractional Edge Lengths LONG BEACH UNIFIED SCHOOL DISTRICT 6 Reposted 11/16/16

7 2-3 I will check my understanding expressions applied to geometry by participating in the FAL. OPTION #2: FORMATIVE ASSESSMENT LESSON Using Space Efficiently: Packing a Truck (SMP 1, 2, 3, 4, 5, 6, 8) I will prepare for the unit assessment on expressions applied to geometry by Incorporating the Standards for Mathematical Practice (SMPs) along with the content standards to review the unit. Ch. 3 Study Help (p. 124) Ch. 3 Review (p ) Ch. 4 Study Help (p. 164) Ch. 4 Review (p ) Ch. 8 Study Help (p. 366) Ch. 8 Review (p ) Geometry Performance Unfolding a Box Unit Assessment Synergy: Unit 2 At this point, all standards addressed in the Math 6 SBAC Interim Assessment Block Geometry have been covered. This block may now be administered. LONG BEACH UNIFIED SCHOOL DISTRICT 7 Reposted 11/16/16