fall 2010 2010 vdoe mathematics institute grades 6-8 focus: patterns, functions, and algebra 2010...

Fall 2010

2010 VDOE Mathematics Institute 2010 VDOE Mathematics Institute

Grades 6-8Grades 6-8Focus: Patterns, Functions, and Algebra Focus: Patterns, Functions, and Algebra

2010 VDOE Mathematics Institute 2010 VDOE Mathematics Institute

Grades 6-8Grades 6-8Focus: Patterns, Functions, and Algebra Focus: Patterns, Functions, and Algebra

Fall 2010

Content Focus Content Focus Content Focus Content Focus Key changes at the middle school level:Key changes at the middle school level:• Properties of Operations with Real Numbers Properties of Operations with Real Numbers • Equations and Expressions Equations and Expressions • InequalitiesInequalities• Modeling Multiplication and Division of Modeling Multiplication and Division of

FractionsFractions• Understanding Mean: Fair Share and Understanding Mean: Fair Share and

Balance PointBalance Point• Modeling Operations with IntegersModeling Operations with Integers

Key changes at the middle school level:Key changes at the middle school level:• Properties of Operations with Real Numbers Properties of Operations with Real Numbers • Equations and Expressions Equations and Expressions • InequalitiesInequalities• Modeling Multiplication and Division of Modeling Multiplication and Division of

FractionsFractions• Understanding Mean: Fair Share and Understanding Mean: Fair Share and

Balance PointBalance Point• Modeling Operations with IntegersModeling Operations with Integers

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Supporting Implementation of Supporting Implementation of 2009 Standards2009 Standards

Supporting Implementation of Supporting Implementation of 2009 Standards2009 Standards

• Highlight key curriculum changes.Highlight key curriculum changes.• Highlight key curriculum changes.Highlight key curriculum changes.

3

• Connect the mathematics across grade levels.Connect the mathematics across grade levels.

• Model instructional strategies.Model instructional strategies.

Fall 2010

Properties of OperationsProperties of OperationsProperties of OperationsProperties of Operations

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Properties of Operations: 2001 StandardsProperties of Operations: 2001 StandardsProperties of Operations: 2001 StandardsProperties 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;

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

5.196.19a

6.19c

6.19b

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Properties of Operations: 2009 StandardsProperties of Operations: 2009 StandardsProperties of Operations: 2009 StandardsProperties of Operations: 2009 Standards

3.20b) Identify examples of the identity and commutative properties for addition and multiplication.

4.16b b) Investigate and describe the associative property for addition and multiplication.

5.19 Investigate and recognize the distributive property of multiplication over addition.

6.19

Investigate and recognizea) the identity properties for addition and multiplication;b) the multiplicative property of zero; andc) the inverse property for multiplication.

7.16

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; ande) the multiplicative property of zero.

8.1aa) simplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers;

8.15c c) identify properties of operations used to solve an equation.

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3.20a&b: Identity Property for Multiplication3.20a&b: Identity Property for Multiplication3.20a&b: Identity Property for Multiplication3.20a&b: Identity Property for Multiplicationx,÷ 1 2 3 4 5 6 7 8 9 10 11 12

1 1 2 3 4 5 6 7 8 9 10 11 12

2 2 4 6 8 10 12 14 16 18 20 22 24

3 3 6 9 12 15 18 21 24 27 30 33 36

4 4 8 12 16 20 24 28 32 36 40 44 48

5 5 10 15 20 25 30 35 40 45 50 55 60

6 6 12 18 24 30 36 42 48 54 60 66 72

7 7 14 21 28 35 42 49 56 63 70 77 84

8 8 16 24 32 40 48 56 64 72 80 88 96

9 9 18 27 36 45 54 63 72 81 90 99 108

10 10 20 30 40 50 60 70 80 90 100 110 120

11 11 22 33 44 55 66 77 88 99 110 121 132

12 12 24 36 48 60 72 84 96 108 120 132 144

The first row and column of products in a

multiplication chart illustrate the identity

property.

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3.20a&b: Commutative Property for Multiplication3.20a&b: Commutative Property for Multiplication3.20a&b: Commutative Property for Multiplication3.20a&b: Commutative Property for Multiplicationx,÷ 1 2 3 4 5 6 7 8 9 10 11 12

1 1 2 3 4 5 6 7 8 9 10 11 12

2 4 6 8 10 12 14 16 18 20 22 24

3 9 12 15 18 21 24 27 30 33 36

4 16 20 24 28 32 36 40 44 48

5 25 30 35 40 45 50 55 60

6 36 42 48 54 60 66 72

7 49 56 63 70 77 84

8 64 72 80 88 96

9 81 90 99 108

10 100 110 120

11 121 132

12 144

Why does the diagonal of perfect squares form a line of symmetry in the chart?

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3.20a&b: Commutative Property for Multiplication3.20a&b: Commutative Property for Multiplication3.20a&b: Commutative Property for Multiplication3.20a&b: Commutative Property for Multiplicationx,÷ 1 2 3 4 5 6 7 8 9 10 11 12

1 1 2 3 4 5 6 7 8 9 10 11 12

2 2 4 6 8 10 12 14 16 18 20 22 24

3 3 6 9 12 15 18 21 24 27 30 33 36

4 4 8 12 16 20 24 28 32 36 40 44 48

5 5 10 15 20 25 30 35 40 45 50 55 60

6 6 12 18 24 30 36 42 48 54 60 66 72

7 7 14 21 28 35 42 49 56 63 70 77 84

8 8 16 24 32 40 48 56 64 72 80 88 96

9 9 18 27 36 45 54 63 72 81 90 99 108

10 10 20 30 40 50 60 70 80 90 100 110 120

11 11 22 33 44 55 66 77 88 99 110 121 132

12 12 24 36 48 60 72 84 96 108 120 132 144

The red rectangle (4x6) and the blue rectangle

(6x4) both cover an area of 24 squares on

the multiplication chart.

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6.19: Multiplicative Property of Zero6.19: Multiplicative Property of Zero6.19: Multiplicative Property of Zero6.19: Multiplicative Property of Zerox,÷ 1 2 3 4 5 6 7 8 9 10 11 12

1 1 2 3 4 5 6 7 8 9 10 11 12

2 2 4 6 8 10 12 14 16 18 20 22 24

3 3 6 9 12 15 18 21 24 27 30 33 36

4 4 8 12 16 20 24 28 32 36 40 44 48

5 5 10 15 20 25 30 35 40 45 50 55 60

6 6 12 18 24 30 36 42 48 54 60 66 72

7 7 14 21 28 35 42 49 56 63 70 77 84

8 8 16 24 32 40 48 56 64 72 80 88 96

9 9 18 27 36 45 54 63 72 81 90 99 108

10 10 20 30 40 50 60 70 80 90 100 110 120

11 11 22 33 44 55 66 77 88 99 110 121 132

12 12 24 36 48 60 72 84 96 108 120 132 144

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.

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Meanings of MultiplicationMeanings of MultiplicationMeanings of MultiplicationMeanings 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.

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3.6: Represent Multiplication Using an Area Model3.6: Represent Multiplication Using an Area Model3.6: Represent Multiplication Using an Area Model3.6: Represent Multiplication Using an Area Model

Use your base ten blocks to represent

3 x 6 = 18

National Library of Virtual Manipulatives – Rectangle Multiplication

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Or did yours look like this?


Commutative Property:

Rotating the rectangle doesn’t change its area.

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Use your base ten blocks to represent

5 x 14 = 70


What is the area of the blue inner rectangle?

What is the area of the red inner rectangle?

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5.19: Distributive Property of Multiplication5.19: Distributive Property of Multiplication5.19: Distributive Property of Multiplication5.19: Distributive Property of Multiplication

How could students record the area of the 5 x 14 rectangle?

5 x 10 = 50

5 x 4 = 20

14 x 5

5 x 10 → 505 x 4 → + 20

70


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5.19: Distributive Property5.19: Distributive Propertyof Multiplication Over Additionof Multiplication Over Addition

5.19: Distributive Property5.19: Distributive Propertyof Multiplication Over Additionof 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.”

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5.19: Distributive Property 5.19: Distributive Property of Multiplication Over Additionof 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.

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The partial products algorithm finds the sum of the areas of four inner rectangles.

Look familiar?F.irst

O.uterI.nnerL.ast

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Strengths of the Area Model of Multiplication Strengths of the Area Model of Multiplication Strengths of the Area Model of 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.

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4.16b: Associative Property for Multiplication4.16b: Associative Property for Multiplication4.16b: Associative Property for Multiplication4.16b: Associative Property for Multiplication

Use your base ten blocks to build a rectangular solid

2cm by 3cm by 4cm

National Library of Virtual Manipulatives – Space Blocks

Base: 2cm by 3cm; Height: 4cmVolume: (2 x 3) x 4 = 24 cm3

Base: 3cm by 4cm; Height: 2cmVolume: 2 x (3 x 4) = 24 cm3

Associative Property: The grouping of the factors does not affect the product.

Fall 2010

Expressions and EquationsExpressions and Equations

Fall 2010

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A Look At Expressions and EquationsA Look At Expressions and Equations

A manipulative, like algebra tiles, A manipulative, like algebra tiles, creates a concrete foundation for creates a concrete foundation for

the abstract, symbolic the abstract, symbolic representations students begin to representations students begin to

wrestle with in middle school.wrestle with in middle school.

Fall 2010

What do these tiles represent?What do these tiles represent?

Tile Bin

1 unit

1 unit Area = 1 square unit

1 unit

Unknown length, x units

Area = x square units

x units

x units

Area = x2 square units

The red tiles denote negative quantities.

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Modeling expressionsModeling expressions

Tile Bin

5 + x

x + 5

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Tile Binx - 1

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Tile Binx + 2

2x

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Tile Binx2 + 3x + 2

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Simplifying expressionsSimplifying expressions

Tile Binx2 + x - 2x2 + 2x - 1

zero pairSimplified expression

-x2 + 3x - 1

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Simplifying expressionsSimplifying expressions

Tile Bin2(2x + 3)

Simplified expression

4x + 6

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Two methods of illustrating the Distributive Property:

Two methods of illustrating the Distributive Property:

Example: 2(2x + 3)

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Fall 2010

Solving Equations

How does this concept progress as we move through middle school?

Solving Equations

How 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 willa) solve one- and two-step linear equations in one variable; andb) solve practical problems requiring the solution of one- and two-step linear

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.

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Solving EquationsSolving Equations

Tile Bin

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Tile Bin x + 3 = 5

6.18 The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions.

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Pictorial Representation:

Symbolic Representation: Condensed Symbolic Representation:

x + 3 = 5

x + 3 = 5 M 3 M 3

x = 2

x + 3 = 5 M 3 M 3

x = 2

Solving EquationsSolving Equations6.18 The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions.

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Tile Bin 2x = 8

Solving EquationsSolving Equations6.18 The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions.

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Fall 2010

Tile Bin 3 = x - 1

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-step linear equations.


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Tile Bin2x + 3 = 13



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x = 5

2x = 10 2 2

2x + 3 = 13 M 3 M 3

2x + 3 = 13

x = 5

2x = 10 2 2

2x + 3 = 13 M 3 M 3

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Tile Bin 0 = 4 – 2x



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0 = 4 – 2x

0 = 4 – 2x̵M 4 M 4

-4 = -2x 2 2

-2 = -x

2 = x

0 = 4 – 2x̵M 4 M 4

-4 = -2x-2 -2

2 = x

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Tile Bin3x + 5 – x = 11

Solving EquationsSolving Equations8.15 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.

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3x + 5 – x = 11

2x + 5 = 11

2x + 5 = 11 -5 -5

2x = 62 2

x = 3

3x + 5 – x = 11

2x + 5 = 11 -5 -5

2x = 62 2

x = 3


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Tile Binx + 2 = 2(2x + 1)


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x + 2 = 2(2x + 1)x + 2 = 4x + 2

x + 2 = 4x + 2-x -x

2 = 3x + 2-2 -2

0 = 3x3 3

0 = x

x + 2 = 2(2x + 1) x + 2 = 4x + 2-x -x

2 = 3x + 2-2 -2

0 = 3x3 3

0 = x

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Fall 2010

Modeling Multiplication and Modeling Multiplication and Division of FractionsDivision of Fractions

Modeling Multiplication and Modeling Multiplication and Division of FractionsDivision of Fractions

45

Fall 2010

So what’s new about fractions So what’s new about fractions in Grades 6-8?in Grades 6-8?

So what’s new about fractions So what’s new about fractions in Grades 6-8?in Grades 6-8?

SOL 6.4 The student will demonstrate multiple representations of multiplication and division of fractions.

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Thinking About MultiplicationThinking About MultiplicationThinking About MultiplicationThinking About Multiplication

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The expression…

We read it… It means… It looks like…

31

2

31

21

32

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Thinking About MultiplicationThinking About MultiplicationThinking About MultiplicationThinking About Multiplication

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The expression…


2 times 3two groups of three

2 timestwo groups of one-third

times one-half group of one-third

31

2

31

21

32

31

31

21

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Making sense of multiplication Making sense of multiplication of fractions using paper of fractions using paper folding and area modelsfolding and area models


Enhanced Scope and Sequence, Enhanced Scope and Sequence, 2004, pages 22 - 242004, pages 22 - 24


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50

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The Importance of ContextThe Importance of Context

• Builds meaning for operations

• Develops understanding of and helps illustrate the relationships among operations

• Allows for a variety of approaches to solving a problem

• Builds meaning for operations

• Develops understanding of and helps illustrate the relationships among operations

• Allows for a variety of approaches to solving a problem

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Fall 2010

Contexts for Modeling Contexts for Modeling Multiplication of FractionsMultiplication of Fractions

Contexts for Modeling Contexts for Modeling Multiplication of FractionsMultiplication of Fractions

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The Andersons had pizza for dinner, and there was one-half of a pizza left over. Their three boys each ate one-third of the leftovers for a late night snack.

How much of the original pizza did each boy get for snack?

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One-third of one-half of a pizza is equal to one-sixth of a pizza.

Which meaning of multiplication does this model fit?

21

31

61

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Andrea and Allison are partners in a relay race. Each girl will run half the total distance. On race day, Andrea stops for water after running of her half of the race.

What portion of the race had Andrea run when she stopped for water?

Andrea and Allison are partners in a relay race. Each girl will run half the total distance. On race day, Andrea stops for water after running of her half of the race.

What portion of the race had Andrea run when she stopped for water?

31

Another Context for Multiplication of Fractions


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Starting Finish

Line Line

½

0 1

Andrea’s half of Allison’s half of

the race the race

Students need experiences with problems that lend themselves to a linear model.

21

31

61

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Mrs. Jones has 24 gold stickers that she bought to put on perfect test papers. She took of the stickers out of the package, and then she used of that half on the papers.

What fraction of the 24 stickers did she use on the perfect test papers?

Mrs. Jones has 24 gold stickers that she bought to put on perfect test papers. She took of the stickers out of the package, and then she used of that half on the papers.

What fraction of the 24 stickers did she use on the perfect test papers?

31

21

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Problems involving discrete items may be represented with set models.Problems involving discrete items may be represented with set models.

What meaning(s) of multiplication does this model fit?

21

31

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One-third of one-half of the 24 stickers is of the 24 stickers.61

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What’s the relationship between What’s the relationship between multiplying and dividing?multiplying and dividing?

What’s the relationship between What’s the relationship between multiplying and dividing?multiplying and dividing?

59

2

1626

Multiplication and division are inverse relations

One operation undoes the other

Division by a number yields the same result as multiplication by its reciprocal (inverse). For example:

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Meanings of DivisionMeanings of DivisionMeanings of DivisionMeanings of Division

For 20 ÷ 5 = 4… Divvy Up (Partitive): “Sally has 20 cookies. How many cookies can she give to each of her five friends, if she gives each friend the same number of cookies?

- Known number of groups, unknown group size Measure Out (Quotitive): “Sally has 20 minutes left on her cell phone plan this month. How many more 5-minute calls can she make this month?

- Known group size, unknown number of groups

Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power, LEA Publishing, 1998.

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Sometimes, Always, Never?Sometimes, Always, Never?Sometimes, Always, Never?Sometimes, Always, Never?

• When we multiply, the product is When we multiply, the product is larger than the number we start with.larger than the number we start with.

• When we divide, the quotient is When we divide, the quotient is smaller than the number we start with.smaller than the number we start with.

• When we multiply, the product is When we multiply, the product is larger than the number we start with.larger than the number we start with.

• When we divide, the quotient is When we divide, the quotient is smaller than the number we start with.smaller than the number we start with.

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““I thought times makes it bigger...”I thought times makes it bigger...”““I thought times makes it bigger...”I thought times makes it bigger...”

When moving beyond whole numbers to situations involving fractions and mixed numbers as factors, divisors, and dividends, students can easily become confused. Helping them match problems to everyday situations can help them better understand what it means to multiply and divide with fractions. However, repeated addition and array meanings of multiplication, as well as a divvy up meaning of division, no longer make as much sense as they did when describing whole number operations.

Using a Groups-Of interpretation of multiplication and a Measure Out interpretation of division can help:

Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power, LEA Publishing, 1998.

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““Groups of” and “Measure Out”Groups of” and “Measure Out”““Groups of” and “Measure Out”Groups of” and “Measure Out”1/4 x 8: “I have one-fourth of a box of 8 doughnuts.” 8 x 1/4: “There are eight quarts of soda on the table. How many whole gallons of soda are there?” 1/2 x 1/3: “The gas tank on my scooter holds 1/3 of a gallon of gas. If I have 1/2 a tank left, what fraction of a gallon of gas do I have in my tank?” 1¼ x 4: “Red Bull comes in packs of four cans. If I have 1¼ packs of Red Bull, how many cans do I have?” 3½ x 2½: “If a cross country race course is 2½ miles long, how many miles have I run after 3½ laps? 3/4 ÷ 2: “How much of a 2-hour movie can you watch in 3/4 of an hour?” *This type may be easier to describe using divvy up. 2 ÷ 3/4: “How many 3/4-of-an-hour videos can you watch in 2 hours?” 3/4 ÷ 1/8: “How many 1/8-sized (of the original pie) pieces of pie can you serve from 3/4 of a pie?” 2½ ÷ 1/3: “A brownie recipe calls for 1/3 of a cup of oil per batch. How many batches can you make if you have 2½ cups of oil left?”

Fall 2010

Thinking About DivisionThinking About DivisionThinking About DivisionThinking About Division

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The expression…


20 ÷ 5

21

20

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65

The expression…


20 ÷ 5 20 divided by 5

20 divided into groups of 5;

20 divided into 5 equal groups…

How many 5’s are in 20?

20 divided by

20 divided into groups of …

How many ’s are in 20?

21

2021

21

21

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The expression… We read it… It means… It looks like…

one-half divided by one-third

divided into

groups of …

How many ’s

are in ?

?31

21

Is the quotient more than one or less than one? How do you know?

21

31

21 3

1

Fall 2010

Contexts for Contexts for Division of FractionsDivision of Fractions

Contexts for Contexts for Division of FractionsDivision of Fractions

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The Andersons had half of a pizza left after dinner. Their son’s typical serving size is pizza. How many of these servings will he eat if he finishes the pizza?

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21

131

21

pizza divided into pizza servings = 1 servings21

21

31

21

serving

1 serving

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Another Context for Another Context for Division of FractionsDivision of FractionsAnother Context for Another Context for Division of FractionsDivision of Fractions

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Marcy is baking brownies. Her recipe calls for cup cocoa for each batch of brownies. Once she gets started, Marcy realizes she only has cup cocoa. If Marcy uses all of the cocoa, how many batches of brownies can she bake?

21

31

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1 cup

cup

0 cups

21 1 batches2

1

One batch (or cup)31

Two batches (or cup)32

Three batches (or cup)33

21

131

21

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Another Context for Another Context for Division of FractionsDivision of Fractions

Mrs. Smith had of a sheet cake left over after her party. She decides to divide the rest of the cake into portions that equal of the original cake.

How many cake portions can Mrs. Smith make from her left-over cake?

31

21

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What could it look like? What could it look like?

31

21

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What does it look What does it look like numerically?like numerically?What does it look What does it look like numerically?like numerically?

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What is the role of common What is the role of common denominators in dividing denominators in dividing

fractions?fractions?

What is the role of common What is the role of common denominators in dividing denominators in dividing

fractions?fractions?

Ensures division of the same size units

Assist with the description of parts of the whole

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What about What about the traditional algorithm?the traditional algorithm?


•If the traditional “invert and multiply” algorithm is taught, it is important that students have the opportunity to consider why it works.•Representations of a pictorial nature provide a visual for finding the reciprocal amount in a given situation.•The common denominator method is a different, valid algorithm. Again, it is important that students have the opportunity to consider why it works.

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Build understanding: Think about 20 ÷ . How many one-half’s are in 20? How many one-half’s are in each of the 20 individual wholes?

Experiences with fraction divisors having a numerator of one illustrate the fact that within each unit, the divisor can be taken out the reciprocal number of times.

21

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Later, think about divisors with numerators > 1. Think about 1 ÷ . How many times could we take from 1? We can take it out once, and we’d have left. We

could only take half of another from the remaining

portion. That’s a total of .

In each unit, there are sets of .

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32

31

32

23

23

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Multiple RepresentationsMultiple RepresentationsMultiple RepresentationsMultiple RepresentationsInstructional programs from pre-k through grade 12 should enable all students to –

•Create and use representations to organize, record and communicate mathematical ideas;

•Select, apply, and translate among mathematical representations to solve problems;

•Use representations to model and interpret physical, social, and mathematical phenomena.

from Principles and Standards for School Mathematics (NCTM, 2000), p. 67.

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Using multiple representations Using multiple representations to express understanding to express understanding

Using multiple representations Using multiple representations to express understanding to express understanding

Given problem

Check your solution

Contextual situation

Solve numerically Solve graphically

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Using multiple representations

to express understanding of division of

fractions

Fall 2010

Mean: Mean: Fair Share and Balance PointFair Share and Balance Point

Mean: Mean: Fair Share and Balance PointFair Share and Balance Point

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Mean: Fair ShareMean: Fair ShareMean: Fair ShareMean: Fair Share

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2009 5.16: The student willa) describe mean, median, and mode as measures of center;b) describe mean as fair share;c) find the mean, median, mode, and range of a set of data; andd) describe the range of a set of data as a measure of variation.

Understanding the Standard: “Mean represents a fair share concept of the data. Dividing the data constitutes a fair share. This is done by equally dividing the data points. This should be demonstrated visually and with manipulatives.”

Fall 2010

Understanding the MeanUnderstanding the MeanUnderstanding the MeanUnderstanding the Mean

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Each person at the table should:

1. Grab one handful of snap cubes.2. Count them and write the number

on a sticky note.3. Snap the cubes together to form a

train.

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84

Work together at your table to answerthe following question:

If you redistributed all of the cubes from your handfuls so that everyone had the same amount (so that they were “shared fairly”), how many cubes would each person receive?

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85

What was your answer?

- How did you handle “leftovers”?

- Add up all of the numbers from the original handfuls and divide the sum by the number of people at the table.

- Did you get the same result?

- What does your answer represent?

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86

Take your sticky note and place it on the wall, so they are ordered…

Horizontally: Low to high, left to right; leave one space if there is a missing number.

Vertically: If your number is already on the wall, place your sticky note in the next open space above that number.

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How did we display our data?

2009 3.17c

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Looking at our line plot, how can we describe our data set? How can we use our line plot to:

- Find the range?

- Find the mode?

- Find the median?

- Find the mean?

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Mean: Balance PointMean: Balance PointMean: Balance PointMean: Balance Point

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2009 6.15: The student willa) describe mean as balance point; andb) decide which measure of center is appropriate for a given purpose.

Essential Knowledge & Skills: • Identify and draw a number line that demonstrates the concept of mean as balance point for a set of data.

Understanding the Standard: “Mean can be defined as the point on a number line where the data distribution is balanced. This means that the sum of the distances from the mean of allthe points above the mean is equal to the sum of the distances of all the data points below the mean.”

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Where is the balance point for this data set?

X

XXX X X

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XXXX X

X

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X

XXX XX

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XXX

XXX

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XXX

X

X

X

3 is theBalance Point

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X

XXX X X

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Fall 20104 is the Balance Point

Move 2 Steps

Move 2 Steps Move 2 Steps

Move 2 Steps


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Fall 2010The Mean is the Balance Point

We can confirm this by calculating:

2 + 2 + 2 + 3 + 3 + 4 + 5 + 7 + 8 = 36

36 ÷ 9 = 4

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The Balance Point is between 10 and 11 (closer to 10).

Move 2 Steps

Move 2 StepsMove 1 Step

Move 1 Step

Where is the balance point for this data set? If we could “zoom in” on the

space between 10 and 11, we could continue this process to arrive at a decimal value for the balance point.

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Mean: Balance PointMean: Balance PointMean: Balance PointMean: Balance Point

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When demonstrating finding the balance point:

1.CHOOSE YOUR DEMONSTRATION DATA SETS INTENTIONALLY.2.Use a line plot to represent the data set.3.Begin with the extreme data points.4.Balance the moves, moving one data point from each side an equal number of steps toward the center.5.Continue until the data is distributed symmetrically or until there are only two values left on the line plot.

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Assessing Higher-Level Assessing Higher-Level ThinkingThinking


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Key Points for 2009 5.16 & 6.15:

Students still need to be able to calculate the mean by summing up and dividing, but they also need to understand:

- why it’s calculated this way (“fair share”);

- how the mean compares to the median and the mode for describing the center of a data set; and

- when each measure of center might be used to represent a data set.

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Mean:Mean:Fair Share & Balance PointFair Share & Balance Point

Mean:Mean:Fair Share & Balance PointFair Share & Balance Point

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“Students need to understand that the mean ‘evens out’ or ‘balances’ a set of data and that the median identifies the ‘middle’ of a data set. They should compare the utility of the mean and the median as measures of center for different data sets. …students often fail to apprehend many subtle aspects of the mean as a measure of center. Thus, the teacher has an important role in providing experiences that help students construct a solid understanding of the mean and its relation to other measures of center.”

- NCTM Principles & Standards for School Mathematics, p. 250

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InequalitiesInequalitiesInequalitiesInequalities

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InequalitiesInequalitiesInequalitiesInequalitiesSOL 6.20 SOL 6.20

The student will graph inequalities on a number line.The student will graph inequalities on a number line.

SOL 7.15 SOL 7.15

The student willThe student will

a)a) solve one-step inequalities in one variable; andsolve one-step inequalities in one variable; and

b)b) graph solutions to inequalities on the number line.graph solutions to inequalities on the number line.

SOL 8.15 SOL 8.15


b)b) solve two-step linear inequalities and graph the solve two-step linear inequalities and graph the results on a number lineresults on a number line

SOL 6.20 SOL 6.20

The student will graph inequalities on a number line.The student will graph inequalities on a number line.

SOL 7.15 SOL 7.15


a)a) solve one-step inequalities in one variable; andsolve one-step inequalities in one variable; and

b)b) graph solutions to inequalities on the number line.graph solutions to inequalities on the number line.

SOL 8.15 SOL 8.15


b)b) solve two-step linear inequalities and graph the solve two-step linear inequalities and graph the results on a number lineresults on a number line

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InequalitiesInequalitiesInequalitiesInequalities

What does inequality mean in the What does inequality mean in the world of mathematics? world of mathematics?

mathematical sentence comparing mathematical sentence comparing two unequal expressionstwo unequal expressions

How are they used in everyday life?How are they used in everyday life?

What does inequality mean in the What does inequality mean in the world of mathematics? world of mathematics?

mathematical sentence comparing mathematical sentence comparing two unequal expressionstwo unequal expressions

How are they used in everyday life?How are they used in everyday life?to solve a problem or describe a relationship for which there is more than one solution

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Equations vs. InequalitiesEquations vs. InequalitiesEquations vs. InequalitiesEquations vs. Inequalities

xx = 2 = 2 x x > 2> 2How are they alike?How are they alike?

How are they different?How are they different?

So, what about So, what about xx >> 2? 2?

xx = 2 = 2 x x > 2> 2How are they alike?How are they alike?


So, what about So, what about xx >> 2? 2?

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xx = 2 = 2

x x > 2> 2

x x >> 2 2

xx = 2 = 2

x x > 2> 2

x x >> 2 2106

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Open or Closed?Open or Closed?Open or Closed?Open or Closed?x > 16x > 16

-5 > y-5 > y

m m >> 12 12

n n << 341 341

-3 < j-3 < j

x > 16x > 16

-5 > y-5 > y

m m >> 12 12

n n << 341 341

-3 < j-3 < j

and, which way should the ray go?and, which way should the ray go?and, which way should the ray go?and, which way should the ray go?107

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x + x + 2 = 82 = 8 x + x + 22 < 8< 8How are they alike?How are they alike?


So, what about So, what about x + x + 2 2 << 8? 8?



So, what about So, what about x + x + 2 2 << 8? 8?

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Both statements include the terms: x, 2 and 8Both statements include the terms: x, 2 and 8

The solution set for both statements involves 6. The solution set for both statements involves 6.

How are they different?How are they different?The solution set for x + 2 = 8 only includes 6. The solution set for The solution set for x + 2 = 8 only includes 6. The solution set for x + x + 22

< 8 does includes all real numbers less than 6. < 8 does includes all real numbers less than 6. What about What about x + x + 2 2 << 8? 8?


Both statements include the terms: x, 2 and 8Both statements include the terms: x, 2 and 8

The solution set for both statements involves 6. The solution set for both statements involves 6.

How are they different?How are they different?The solution set for x + 2 = 8 only includes 6. The solution set for The solution set for x + 2 = 8 only includes 6. The solution set for x + x + 22

< 8 does includes all real numbers less than 6. < 8 does includes all real numbers less than 6. What about What about x + x + 2 2 << 8? 8?

The solution set for this inequality includes 6 and The solution set for this inequality includes 6 and all real numbers less than 6. all real numbers less than 6.

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x+x+ 2 = 8 2 = 8

x+x+ 2 < 8 2 < 8

x+x+ 2 2 << 8 8

x+x+ 2 = 8 2 = 8

x+x+ 2 < 8 2 < 8

x+x+ 2 2 << 8 8

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Inequality MatchInequality MatchInequality MatchInequality Match

With your tablemates, find as With your tablemates, find as many matches as possible in many matches as possible in

the set of cards.the set of cards.

With your tablemates, find as With your tablemates, find as many matches as possible in many matches as possible in

the set of cards.the set of cards.

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X > 5 X is greaterthan 5

SAMPLE MATCH112

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Operations with IntegersOperations with IntegersOperations with IntegersOperations with Integers

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Operations with IntegersOperations with IntegersOperations with IntegersOperations with Integers2009 7.3a: The student will

a) model addition, subtraction, multiplication and division of integers; andb) add, subtract, multiply, and divide integers.

Is this really a “new” SOL?

2001 7.5: The student will formulate rules for and solve practical problems involving basic operations (addition, subtraction, multiplication, and division) with integers.

“Model”114

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7.3a: The student will model addition, subtraction, multiplication, and division of integers.

What operation does this model?

= 1 = -1

3 + (-7) = -4115

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= 1 = -1

3 • (-4) = -12116

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5 + (-17) = -125 - 17 = -12

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3 • (-5) = -15

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Another Example Another Example of Assessing of Assessing Higher-Level ThinkingHigher-Level Thinking

Another Example Another Example of Assessing of Assessing Higher-Level ThinkingHigher-Level Thinking

7.5c: The student will describe how changing one measured attribute of a rectangular prism affects its volume and surface area.

Describe how the volume of the rectangular prism shown (height = 8 in.) would be affected if the height was increased by a scale factor of ½ or 2.

8 in.

5 in.3 in.

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Tying it All TogetherTying it All TogetherTying it All TogetherTying it All Together

1.1. Improved vertical alignment of Improved vertical alignment of content with increased cognitive content with increased cognitive demand.demand.

2.2. Key conceptual models can be Key conceptual models can be extended across grade levels.extended across grade levels.

3. Refer to the Curriculum Framework.

4.4. Pay attention to the changes in the Pay attention to the changes in the verbs.verbs.

1.1. Improved vertical alignment of Improved vertical alignment of content with increased cognitive content with increased cognitive demand.demand.

2.2. Key conceptual models can be Key conceptual models can be extended across grade levels.extended across grade levels.

3. Refer to the Curriculum Framework.

4.4. Pay attention to the changes in the Pay attention to the changes in the verbs.verbs.

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Exit Slip Exit Slip Exit Slip Exit Slip

1. Aha... 1. Aha...

2. Can’t wait to share…2. Can’t wait to share…

3. HELP!3. HELP!

1. Aha... 1. Aha...

2. Can’t wait to share…2. Can’t wait to share…

3. HELP!3. HELP!

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fall 2010 2010 vdoe mathematics institute grades 6-8 focus: patterns, functions, and algebra 2010...

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