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1. 2010 VDOE Mathematics Institute • Grades 6-8 • Focus: Patterns, Functions, and Algebra

2. 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

3. Supporting Implementation of 2009 Standards • Highlight key curriculum changes. • Connect the mathematics across grade levels. • Model instructional strategies.

4. Properties of Operations

5. 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

6. Properties of Operations: 2009 Standards

7. 3.20a&b: Identity Property for Multiplication The first row and column of products in a multiplication chart illustrate the identity property.

8. 3.20a&b: Commutative Property for Multiplication Why does the diagonal of perfect squares form a line of symmetry in the chart?

9. 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.

10. 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.

11. 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.

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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.

20. 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

21. Expressions and Equations

22. 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

23. 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.

24. Modeling expressions Tile Bin x + 5 5 + x

25. Modeling expressions Tile Bin x - 1

26. Modeling expressions Tile Bin x + 2 2x

27. Modeling expressions Tile Bin x2 + 3x + 2

28. Simplifying expressions Tile Bin x2 + x - 2x2 + 2x - 1 zero pair Simplified expression -x2 + 3x - 1

29. Simplifying expressions Tile Bin 2(2x + 3) Simplified expression 4x + 6

30. Two methods of illustrating the Distributive Property: Example: 2(2x + 3)

31. 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.

32. Solving Equations Tile Bin

33. 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

34. 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.

35. 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

36. 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

37. 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

38. 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.

39. 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

40. 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.

41. 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

42. 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.

43. 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)

44. 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.

45. 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.

46. Thinking About Multiplication

47. Making sense of multiplication of fractions using paper folding and area models Enhanced Scope and Sequence, 2004, pages 22 - 24

48. Making sense of multiplication of fractions using paper folding and area models Enhanced Scope and Sequence, 2004, pages 22 - 24