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2010 VDOE Mathematics Institute Grades 6-8 Focus: Patterns, Functions, and Algebra

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## 2010 VDOE Mathematics Institute Grades 6-8 Focus: Patterns, Functions, and Algebra

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**2010 VDOE Mathematics Institute**• Grades 6-8 • Focus: Patterns, Functions, and Algebra**Content Focus**• Key changes at the middle school level: • Properties of Operations with Real Numbers • Equations and Expressions • Inequalities • Modeling Multiplication and Division of Fractions • Understanding Mean: Fair Share and Balance Point • Modeling Operations with Integers**Supporting Implementation of 2009 Standards**• Highlight key curriculum changes. • Connect the mathematics across grade levels. • Model instructional strategies.**Properties of Operations: 2001 Standards**• 7.3 The student will identify and apply the following properties of • operations with real numbers: • a) the commutative and associative properties for addition and • multiplication; • b) the distributive property; • c) the additive and multiplicative identity properties; • d) the additive and multiplicative inverse properties; and • e) the multiplicative property of zero. • 8.1 The student will • a) simplify numerical expressions involving positive exponents, • using rational numbers, order of operations, and properties of • operations with real numbers; 3.20a&b; 4.16b 5.19 6.19a 6.19c 6.19b**3.20a&b: Identity Property for Multiplication**The first row and column of products in a multiplication chart illustrate the identity property.**3.20a&b: Commutative Property for Multiplication**Why does the diagonal of perfect squares form a line of symmetry in the chart?**3.20a&b: Commutative Property for Multiplication**The red rectangle (4x6) and the blue rectangle (6x4) both cover an area of 24 squares on the multiplication chart.**6.19: Multiplicative Property of Zero**6 x 0 = 0 0 x 6 = 0 Area multiplication is based on rectangles. If one factor is zero, then the number sentence doesn’t describe a rectangle, it describes a line segment, and the product (the “area”) is zero.**Meanings of Multiplication**For 5 x 4 = 20… Repeated Addition: “4, 8, 12, 16, 20.” Groups-Of: “Five bags of candy with four pieces of candy in each bag.” Rectangular Array: “Five rows of desks with four desks in each row.” Rate: “Dave bought five raffle tickets at $4.00 apiece.” or “Dave walked four miles per hour for five hours.” Comparison: “Alice has 4 cookies; Ralph has five times as many.” Combinations: “Cindy has five different shirts and four different pairs of pants; how many different shirt/pants outfits can she make?” Area: “Ricky buys a rectangular rug 5 feet long and 4 feet wide.” Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power, LEA Publishing, 1998, Chapter 5.**3.6: Represent Multiplication Using an Area Model**Use your base ten blocks to represent 3 x 6 = 18 National Library of Virtual Manipulatives – Rectangle Multiplication**3.6: Represent Multiplication Using an Area Model**Or did yours look like this? Rotating the rectangle doesn’t change its area. Commutative Property: National Library of Virtual Manipulatives – Rectangle Multiplication**3.6: Represent Multiplication Using an Area Model**Use your base ten blocks to represent 5 x 14 = 70 What is the area of the red inner rectangle? What is the area of the blue inner rectangle? National Library of Virtual Manipulatives – Rectangle Multiplication**3.6: Represent Multiplication Using an Area Model**5.19: Distributive Property of Multiplication How could students record the area of the 5 x 14 rectangle? 5 x 4 = 20 14 x 5 5 x 10 → 50 5 x 4 → + 20 70 5 x 10 = 50**5.19: Distributive Propertyof Multiplication Over Addition**Understanding the Standard: “The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products (e.g., 3(4 + 5) = 3 x 4 + 3 x 5, 5 x (3 + 7) = (5 x 3) + (5 x 7); or (2 x 3) + (2 x 5) = 2 x (3 + 5).” • Essential Knowledge & Skills: • “Investigate and recognize the distributive property of whole • numbers, limited to multiplication over addition, using • diagrams and manipulatives.” • “Investigate and recognize an equation that represents the • distributive property, when given several whole number • equations, limited to multiplication over addition.” National Library of Virtual Manipulatives – Rectangle Multiplication**5.19: Distributive Property of Multiplication Over Addition**Use base ten blocks to build a 12 x 23 rectangle. The traditional multi-digit multiplication algorithm finds the sum of the areas of two inner rectangles. National Library of Virtual Manipulatives – Rectangle Multiplication**5.19: Distributive Property of Multiplication Over Addition**The partial products algorithm finds the sum of the areas of four inner rectangles. Look familiar? F.irst O.uter I.nner L.ast National Library of Virtual Manipulatives – Rectangle Multiplication**Strengths of the Area Model of Multiplication**• Illustrates the inherent connections between multiplication and division: • Factors, divisors, and quotients are represented by the • lengths of the rectangle’s sides. • Products and dividends are represented by the area of • the rectangle. • Versatile: • Can be used with whole numbers and decimals (through • hundredths). • Rotating the rectangle illustrates commutative property. • Forms the basis for future modeling: distributive • property; factoring with Algebra Tiles; and Completing • the Square to solve quadratic equations.**4.16b: Associative Property for Multiplication**Use your base ten blocks to build a rectangular solid 2cm by 3cm by 4cm Base: 3cm by 4cm; Height: 2cm Volume: 2 x (3 x 4) = 24 cm3 Associative Property: The grouping of the factors does not affect the product. Base: 2cm by 3cm; Height: 4cm Volume: (2 x 3) x 4 = 24 cm3 National Library of Virtual Manipulatives – Space Blocks**A Look At Expressions and Equations**A manipulative, like algebra tiles, creates a concrete foundation for the abstract, symbolic representations students begin to wrestle with in middle school. 22**What do these tiles represent?**1 unit Tile Bin 1 unit Area = 1 square unit Unknown length, x units Area = x square units 1 unit x units x units Area = x2 square units The red tiles denote negative quantities.**Modeling expressions**Tile Bin x + 5 5 + x**Modeling expressions**Tile Bin x - 1**Modeling expressions**Tile Bin x + 2 2x**Modeling expressions**Tile Bin x2 + 3x + 2**Simplifying expressions**Tile Bin x2 + x - 2x2 + 2x - 1 zero pair Simplified expression -x2 + 3x - 1**Simplifying expressions**Tile Bin 2(2x + 3) Simplified expression 4x + 6**Two methods of illustrating the Distributive Property:**Example: 2(2x + 3)**Solving EquationsHow does this concept progress as we move**through middle school? 6th grade: 6.18 The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions. • 7th grade: • 7.14 The student will • solve one- and two-step linear equations in one variable; and • solve practical problems requiring the solution of one- and two-steplinear equations. 8th grade: 8.15 The student will a) solve multistep linear equations in one variable on one and two sides of the equation; b) solve two-step linear inequalities and graph the results on a number line; and c) identify properties of operations used to solve an equation.**Solving Equations**Tile Bin**Solving Equations**Tile Bin 6.18 The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions. x + 3 = 5**x + 3 = 5**x + 3 = 5 ̵ 3 ̵ 3 x = 2 x + 3 = 5 ̵ 3 ̵ 3 x = 2 Solving Equations 6.18 The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions.**Tile Bin**Solving Equations 6.18 The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions. 2x = 8**Tile Bin**Solving Equations 7.14 The student will solve one- and two-step linear equations in one variable; and solve practical problems requiring the solution of one- and two-steplinear equations. 3 = x - 1**Tile Bin**Solving Equations 7.14 The student will solve one- and two-step linear equations in one variable; and solve practical problems requiring the solution of one- and two-steplinear equations. 2x + 3 = 13**Solving Equations**2x + 3 = 13 2x + 3 = 13 ̵ 3 ̵ 3 2x + 3 = 13 ̵ 3 ̵ 3 2x= 10 2 2 2x= 10 2 2 x = 5 x = 5 7.14 The student will solve one- and two-step linear equations in one variable; and solve practical problems requiring the solution of one- and two-steplinear equations.**Solving Equations**Tile Bin 7.14 The student will solve one- and two-step linear equations in one variable; and solve practical problems requiring the solution of one- and two-steplinear equations. 0 = 4 – 2x**Solving Equations**0 = 4 – 2x 0 = 4 – 2x ̵ 4 ̵ 4 0 = 4 – 2x ̵ 4 ̵ 4 -4= -2x -2 -2 -4= -2x 2 2 2 = x -2 = -x 2 = x 7.14 The student will solve one- and two-step linear equations in one variable; and solve practical problems requiring the solution of one- and two-steplinear equations.**Tile Bin**Solving Equations 8.15 The student will a) solve multistep linear equations in one variable on one and two sides of the equation; b) solve two-step linear inequalities and graph the results on a number line; and c) identify properties of operations used to solve an equation. 3x + 5 – x = 11**Solving Equations**3x + 5 – x = 11 3x + 5 – x = 11 2x + 5 = 11 2x + 5 = 11 -5 -5 2x + 5 = 11 -5 -5 2x = 6 2 2 2x = 6 2 2 x = 3 x = 3 8.15 The student will a) solve multistep linear equations in one variable on one and two sides of the equation; b) solve two-step linear inequalities and graph the results on a number line; and c) identify properties of operations used to solve an equation.**Tile Bin**Solving Equations 8.15 The student will a) solve multistep linear equations in one variable on one and two sides of the equation; b) solve two-step linear inequalities and graph the results on a number line; and c) identify properties of operations used to solve an equation. x + 2 = 2(2x + 1)**Solving Equations**x + 2 = 2(2x + 1) x + 2 = 4x + 2 x + 2 = 2(2x + 1) x + 2 = 4x + 2 -x -x x + 2 = 4x + 2 -x -x 2 = 3x + 2 -2 -2 2 = 3x + 2 -2 -2 0 = 3x 3 3 0 = 3x 3 3 0 = x 0 = x 8.15 The student will a) solve multistep linear equations in one variable on one and two sides of the equation; b) solve two-step linear inequalities and graph the results on a number line; and c) identify properties of operations used to solve an equation.**So what’s new about fractions in Grades 6-8?**SOL 6.4 The student will demonstrate multiple representations of multiplication and division of fractions.**Making sense of multiplication of fractions using paper**folding and area models Enhanced Scope and Sequence, 2004, pages 22 - 24**Making sense of multiplication of fractions using paper**folding and area models Enhanced Scope and Sequence, 2004, pages 22 - 24