unpacking the middle school mathematics curriculum august 16, 2011

Unpacking the Middle School Mathematics Curriculum August 16, 2011

2009 Mathematics Standards of Learning 2 Rigor has been increased Repetition has been decreased Retention and application of content from previous years required Vertical alignment has been improved

Enhanced Scope and Sequence Lesson Plans 3 Revised and redeveloped New layout Provides differentiation strategies for all types of learners Anticipated by Summer 2011

Blueprints and Curriculum Blueprints are currently available on the VDOE website to accommodate curriculum development and instructional planning, but will not become effective until the 2011-2012 school year School divisions should be teaching the new content from the 2009 SOL in the 2010-2011 school year since there will be FT items in spring 2011 on the new content - 4 -

New SOL Blueprints Look for changes in: Number of reporting categories Number of items in reporting categories Asterisks denoting SOL that will be assessed in the non-calculator section for grades 4 - 7 SOL that will not be tested 5

Grades 6-8 Reporting Categories 7 Reporting Categories6 (2001)6 (2009)7 (2001)7 (2009)8 (2001)8 (2009)20012009 Number and Number Sense 8107 16 7 14 15% 33% Computation and Estimation 1097716% Measurement and Geometry 12 13121424%26% Probability and Statistics 8 19 12 21 8 22 19% 41% Patterns, Functions, and Algebra 12 1627% Total Questions 50 100%

Formula Sheets Formula sheets that correspond to the 2009 Standards for grades 6-8 and EOC are currently available on the VDOE 2011-2012 Ancillary Test Materials webpage http://www.doe.virginia.gov/testing/test_administration/a ncilliary_materials/2011-12/index.shtml - 8 -

Vertical Articulation Documents 9 Click here for documents

Vertical Articulation of Content 10 Consistency Connections Relevance The Mathematics Crosswalk Between the 2009 and 2001 Standards (PDF) provides detail on additions, deletions and changes included in the 2009 Mathematics Standards of Learning.Mathematics Crosswalk Between the 2009 and 2001 Standards Why is it important knowledge to have? All these lead to deeper understanding and long-term retention of content

Examine the 5-8 Vertical Articulation 11 Identify the similarities and differences between the grade levels What are the key verbs? Was there anything that surprised you? Breaking Down the 6-8 Standards List the 5 most important concepts you see in Grades 6 and 7 Can you draw a representation of the topics?

Number and Number Sense & Computation and Estimation 12 Grade 6

Measurement and Geometry 15 Grade 6

Probability, Statistics, Patterns, Functions, and Algebra 18 Grade 6

Todays Content Focus Key changes at the middle school level: 1. Properties of Operations with Real Numbers 2. Equations and Expressions 3. Inequalities 4. Modeling Multiplication and Division of Fractions 5. Understanding Mean: Fair Share and Balance Point 6. Modeling Operations with Integers 21

Supporting Implementation of 2009 Standards Highlight key curriculum changes. Connect the mathematics across grade levels. Model instructional strategies. 22

Properties of Operations 23

24 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

25 Properties of Operations: 2009 Standards 3.20 b) Identify examples of the identity and commutative properties for addition and multiplication. 4.16bb) Investigate and describe the associative property for addition and multiplication. 5.19Investigate and recognize the distributive property of multiplication over addition. 6.19 Investigate and recognize a) the identity properties for addition and multiplication; b) the multiplicative property of zero; and c) 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; and e) the multiplicative property of zero. 8.1a a) simplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers; 8.15cc) identify properties of operations used to solve an equation.

26 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 Childrens Mathematical Power, LEA Publishing, 1998, Chapter 5.

Multiplication and Area (Grade 3) 27 2 groups of 3 2 x 3 = 6 2 x 3 Concept of multiplication Connection to area Area is 6 square units 2 3

28 Represent Multiplication Using an Area Model (SOL 3.6) 3 x 6 = 18 National Library of Virtual Manipulatives Rectangle Multiplication

29 Or does it look like this? National Library of Virtual Manipulatives Rectangle Multiplication Commutative Property: Rotating the rectangle doesnt change its area. Represent Multiplication Using an Area Model (SOL 3.6)

30 Associative Property for Multiplication (SOL 4.16b) 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: 4cm Volume: (2 x 3) x 4 = 24 cm 3 Base: 3cm by 4cm; Height: 2cm Volume: 2 x (3 x 4) = 24 cm 3 Associative Property: The grouping of the factors does not affect the product.

Multiplication and Area (Grade 3 and 4) 31 Multiplying whole numbers progression of complexity 8 x 10 10 8 12 23 8 groups of 10

Multiplication and Area (Grade 3 and 4) 32 Multiplying whole numbers 12 23 20 3 10 2 Partial Products

Multiplication and Area (Algebra I) 33 Connection to Algebra I x 3 2 x This will work for more than multiplying binomials! (unlike FOIL). This model is directly linked to use of algebra tiles.

Multiplication and Area (Algebra I/Geometry Application) 34 original warehouse x x 3 2 The sides of a square warehouse are increased by 2m and 3m as shown. The area of the extended warehouse is 156 m 2. What was the side length of the original warehouse? New Zealand Level 1 Algebra 1 Asia-Pacific Economic Cooperation Mathematics Assessment DatabaseMathematics Assessment Database

Multiplication and Area (Algebra I/Geometry Application) 35 original warehouse 30 50 x x The original warehouse measured 30m by 50m. The owner would like to know the smallest length by which she would need to extend each side in order to have a total area of 2500 m 2. New Zealand Level 1 Algebra 1 (modified) Asia-Pacific Economic Cooperation Mathematics Assessment DatabaseMathematics Assessment Database

36 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 rectangles 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.

Expressions and Equations

38 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.

What do these tiles represent? Tile Bin 1 unit Area = 1 square unit 1 unit Unknown length, x units Area = x square units x units Area = x 2 square units The red tiles denote negative quantities. 39

Modeling expressions Tile Bin 5 + x x + 5 40

Modeling expressions Tile Bin x - 1 41

Modeling expressions Tile Bin x + 2 2x2x 42

Modeling expressions Tile Bin x 2 + 3x + 2 43

Simplifying expressions Tile Bin x 2 + x - 2x 2 + 2x - 1 zero pair Simplified expression -x 2 + 3x - 1 44

Simplifying expressions Tile Bin 2(2x + 3) Simplified expression 4x + 6 45

Two methods of illustrating the Distributive Property: Example: 2(2x + 3) 46

Solving Equations How does this concept progress as we move through middle school? 6 th grade: 6.18 The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions. 7 th grade: 7.14The student will a)solve one- and two-step linear equations in one variable; and b)solve practical problems requiring the solution of one- and two-step linear equations. 8 th 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. ** What does this mean for Course 2 students who go to Algebra 1? 47

Solving Equations Tile Bin 48

Solving Equations 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. 49

Pictorial Representation:Symbolic Representation:Condensed Symbolic Representation: x + 3 = 5 3 3 x = 2 x + 3 = 5 3 3 x = 2 6.18 The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions. 50 Solving Equations

Tile Bin 2x = 8 6.18 The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions. 51 Solving Equations

Tile Bin 3 = x - 1 7.14The 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. 52 Solving Equations

Tile Bin 2x + 3 = 13 7.14The 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. 53 Solving Equations

7.14The 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. Pictorial Representation:Symbolic Representation:Condensed Symbolic Representation: x = 5 2x = 10 2 2 2x + 3 = 13 3 3 2x + 3 = 13 x = 5 2x = 10 2 2 2x + 3 = 13 3 3 54 Solving Equations

Tile Bin 0 = 4 2x 7.14The 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. 55 Solving Equations

7.14The 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. Pictorial Representation:Symbolic Representation:Condensed Symbolic Representation: 0 = 4 2x 4 -4 = -2x 2 2 -2 = -x 2 = x 0 = 4 2x 4 -4 = -2x -2 2 = x 56 Solving Equations

Tile Bin 3x + 5 x = 11 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. 57 Solving Equations

Pictorial Representation:Symbolic Representation:Condensed Symbolic Representation: 3x + 5 x = 11 2x + 5 = 11 -5 -5 2x = 6 2 x = 3 3x + 5 x = 11 2x + 5 = 11 -5 -5 2x = 6 2 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. 58 Solving Equations

Tile Bin x + 2 = 2(2x + 1) 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. 59 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. Pictorial Representation:Symbolic Representation:Condensed Symbolic Representation: x + 2 = 2(2x + 1) x + 2 = 4x + 2 -x 2 = 3x + 2 -2 0 = 3x 3 0 = x x + 2 = 2(2x + 1) x + 2 = 4x + 2 -x 2 = 3x + 2 -2 0 = 3x 3 0 = x 60 Solving Equations

Inequalities 61

SOL 6.20 The student will graph inequalities on a number line. SOL 7.15 The student will a)solve one-step inequalities in one variable; and graph solutions to inequalities on the number line. SOL 8.15 The student will b)solve two-step linear inequalities and graph the results on a number line 62 Inequalities

What does inequality mean in the world of mathematics? mathematical sentence comparing two unequal expressions How are they used in everyday life? to solve a problem or describe a relationship for which there is more than one solution 63

Equations vs. Inequalities x = 2x > 2 How are they alike? How are they different? So, what about x > 2? 64

Equations vs. Inequalities x = 2 x > 2 65

Open or Closed? x > 16 -5 > y m > 12 n < 341 -3 < j and, which way should the ray go? 66

Equations vs. Inequalities x + 2 = 8 x + 2 < 8 How are they alike? How are they different? So, what about x + 2 < 8? 67

Equations vs. Inequalities x + 2 = 8 x + 2 < 8 How are they alike? Both statements include the terms: x, 2 and 8 The solution set for both statements involves 6. How are they different? The solution set for x + 2 = 8 only includes 6. The solution set for x + 2 < 8 does includes all real numbers less than 6. What about x + 2 < 8? The solution set for this inequality includes 6 and all real numbers less than 6. 68

Equations vs. Inequalities x+ 2 = 8 x+ 2 < 8 69

Inequality Match Classroom activity: With your tablemates, find as many matches as possible in the set of cards. Tidewater Team: Inequality Match CardsInequality Match Cards 70

X > 5 X is greater than 5 SAMPLE MATCH 71

Modeling Multiplication and Division of Fractions 72

So whats new about fractions in Grades 6-8? SOL 6.4 The student will demonstrate multiple representations of multiplication and division of fractions. 73

Thinking About Multiplication 74 The expression We read itIt meansIt looks like

Thinking About Multiplication 75 The expression We read itIt meansIt looks like 2 times 3 two groups of three 2 times two groups of one-third timesone-half group of one-third

Making sense of multiplication of fractions using paper folding and area models Enhanced Scope and Sequence, 2004, pages 22 - 24 76

The 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 77

Contexts for Modeling Multiplication of Fractions 78 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?

79 One-third of one-half of a pizza is equal to one-sixth of a pizza. Which meaning of multiplication does this model fit?

Another Context for Multiplication of Fractions 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? 80

Problems involving discrete items may be represented with set models. What meaning(s) of multiplication does this model fit? One-third of one-half of the 24 stickers is of the 24 stickers. 81

Whats the relationship between multiplying and dividing? 82 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:

83 Meanings 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 Childrens Mathematical Power, LEA Publishing, 1998.

Sometimes, Always, Never? When we multiply, the product is larger than the number we start with. When we divide, the quotient is smaller than the number we start with. 84

85 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 Childrens Mathematical Power, LEA Publishing, 1998.

86 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? 1 x 4: Red Bull comes in packs of four cans. If I have 1 packs of Red Bull, how many cans do I have? 3 x 2: If a cross country race course is 2 miles long, how many miles have I run after 3 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? 2 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 2 cups of oil left?

Thinking About Division 87 The expression We read itIt meansIt looks like 20 5

88 The expression We read itIt meansIt looks like 20 520 divided by 5 20 divided into groups of 5; 20 divided into 5 equal groups How many 5s are in 20? 20 divided by 20 divided into groups of How many s are in 20? 88 Thinking About Division

89 The expressionWe read itIt meansIt looks like one-half divided by one-third divided into groups of How many s are in ? ? Is the quotient more than one or less than one? How do you know? Thinking About Division

Contexts for Division of Fractions 90 The Andersons had half of a pizza left after dinner. Their sons typical serving size is pizza. How many of these servings will he eat if he finishes the pizza?

pizza divided into pizza servings = 1 servingsserving 1 serving 91

Another Context for Division of Fractions 92 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?

1 cup cup 0 cups 1 batches One batch (or cup) Two batches (or cup) Three batches (or cup) 93

94 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? Another Context for Division of Fractions

95 What could it look like?

What does it look like numerically? 96

What is the role of common denominators in dividing fractions? Ensures division of the same size units Assist with the description of parts of the whole 97

98 What about the traditional algorithm? What about 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.

99 Build understanding: Think about 20 . How many one-halfs are in 20? How many one-halfs 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. What about the traditional algorithm? What about the traditional algorithm?

100 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 wed have left. We could only take half of another from the remaining portion. Thats a total of. In each unit, there are sets of. What about the traditional algorithm? What about the traditional algorithm?

Multiple Representations Instructional 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. 101

Using multiple representations to express understanding Given problem Check your solution Contextual situation Solve numericallySolve graphically 102

103 Using multiple representations to express understanding of division of fractions

Mean: Fair Share and Balance Point 104

Mean: Fair Share 105 2009 5.16: The student will a)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; and d)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.

Understanding the Mean 106 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.

Understanding the Mean 107 Work together at your table to answer the 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?

Understanding the Mean 108 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?

Understanding the Mean 109 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.

Understanding the Mean 110 How did we display our data? 2009 3.17c

Understanding the Mean 111 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?

Mean: Balance Point 112 2009 6.15: The student will a) describe mean as balance point; and b) 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 all the points above the mean is equal to the sum of the distances of all the data points below the mean.

Where is the balance point for this data set? X X X XXX 113

Where is the balance point for this data set? X X X XX X 114

Where is the balance point for this data set? X XX X X X 116

Where is the balance point for this data set? X X X X X X 3 is the Balance Point 117

4 is the Balance Point Move 2 Steps Where is the balance point for this data set? 119

The Mean is the Balance Point We can confirm this by calculating: 2 + 2 + 2 + 3 + 3 + 4 + 5 + 7 + 8 = 36 36 9 = 4 120

The Balance Point is between 10 and 11 (closer to 10). Move 2 Steps 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. 121

Mean: Balance Point 122 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.

Assessing Higher-Level Thinking 123 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 its 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.

Mean: Fair Share & Balance Point 124 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

Operations with Integers 125

Operations with Integers 2009 7.3a: The student will a) model addition, subtraction, multiplication and division of integers; and b) 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 126

Assessing Higher-Level Thinking 7.3a: The student will model addition, subtraction, multiplication, and division of integers. What operation does this model? = 1 = -1 3 + (-7) = -4 127

Assessing Higher-Level Thinking 7.3a: The student will model addition, subtraction, multiplication, and division of integers. What operation does this model? = 1 = -1 3 (-4) = -12 128

Assessing Higher-Level Thinking 7.3a: The student will model addition, subtraction, multiplication, and division of integers. What operation does this model? 5 + (-17) = -125 - 17 = -12 129

Assessing Higher-Level Thinking 7.3a: The student will model addition, subtraction, multiplication, and division of integers. What operation does this model? 3 (-5) = -15 130

Another Example of Assessing Higher- 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 or 2. 8 in. 5 in. 3 in. 131

Tying it All Together 1. Improved vertical alignment of content with increased cognitive demand. 2. Key conceptual models can be extended across grade levels. 3. Refer to the Curriculum Framework. 4. Pay attention to the changes in the verbs. 132

Narrowing Achievement Gaps 133 Ask high-level questions of all students Consistently provide multiple representations Facilitate connections Solicit multiple student solutions Engage students in the learning process

Narrowing Achievement Gaps 134 Promote mathematical communication Listen carefully to your students words and learn from them Provide immediate feedback on all work Give students challenging but accessible tasks

No Pain, No Gain 135 Pertains to learning mathematics as well Let kids struggle to make sense of the mathematics

VDOE Resources 136 Technical Assistance Documents for SOL A.9 and SOL AII.11SOL A.9 and SOL AII.11 Mathematics Institutes: Training/instructional resources for K-Algebra II available through the Tidewater Team at William and Mary Website Tidewater Team at William and Mary Website

unpacking the middle school mathematics curriculum august 16, 2011

Documents

improved slide

number sense computation

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new sol blueprints

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mathematics crosswalk

standards pdf