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2010 VDOE Mathematics Institute Grades 6-8 Focus: Patterns, Functions, and Algebra Fall 2010 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 Fall 2010 2 Supporting Implementation of 2009 Standards • Highlight key curriculum changes. • Connect the mathematics across grade levels. • Model instructional strategies. Fall 2010 3 Properties of Operations Fall 2010 4 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; 3.20a&b; 4.16b b) the distributive property; 5.19 c) the additive and multiplicative identity properties; 6.19a d) the additive and multiplicative inverse properties; and e) the multiplicative property of zero. 6.19c 6.19b 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; Fall 2010 5 Properties of Operations: 2009 Standards 3.20 b) 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 6.19 7.16 8.1a Investigate and recognize the distributive property of multiplication over addition. Investigate and recognize a) the identity properties for addition and multiplication; b) the multiplicative property of zero; and c) the inverse property for multiplication. 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. a) 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. Fall 2010 6 3.20a&b: Identity Property for Multiplication Fall 2010 x,÷ 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 9 9 18 27 36 10 10 20 30 40 11 11 22 33 44 12 12 24 36 48 60 72 84 64 72 80 88 96 The first row and 45 54 63 of72products 81 90 99in108 column a chart 50 60 multiplication 70 80 90 100 110 120 illustrate the identity 55 66 77 88 99 110 121 132 property. 96 108 120 132 144 7 3.20a&b: Commutative Property for Multiplication x,÷ 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 4 6 8 10 12 14 16 18 20 22 24 9 12 15 18 21 24 27 30 33 36 16 20 24 28 32 36 40 44 48 25 30 35 40 45 50 55 60 36 42 48 54 60 66 72 49 56 63 70 77 84 64 72 80 88 96 81 90 99 108 100 110 120 121 132 2 3 4 5 6 7 8 Why does the 9 diagonal of perfect 10 squares form a line of 11 symmetry in the chart? 12 Fall 2010 144 8 3.20a&b: Commutative Property for Multiplication Fall 2010 x,÷ 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 9 9 18 27 36 45 10 10 20 30 40 50 11 11 22 33 44 55 12 12 24 36 48 60 64 red 72 rectangle 80 88 96 The 54 (4x6) 63 72 and 81 90the99blue 108 rectangle (6x4)110both 60 70 80 90 100 120 cover an area of 24 66 77 88 99 110 121 132 squares on the 72 84 96 108 120 132 144 multiplication chart. 56 9 6.19: Multiplicative Property of Zero x,÷ 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 10 10 11 11 12 12 Fall 2010 Area multiplication is based on 81 90 99 108 rectangles. If one factor is 20 30 40 50 60 70 80 90 100 110 120 zero, then the number sentence 22 33 44 55 66 77 88 99 110 121 132 doesn’t describe a rectangle, it 24 36 48 60 72 84 96 108 120 132 144 describes a line segment, and the product (the “area”) is zero. 10 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. Fall 2010 11 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 Fall 2010 12 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 Fall 2010 13 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 Fall 2010 14 3.6:5.19: Represent Distributive Multiplication Property Using of Multiplication an Area Model How could students record the area of the 5 x 14 rectangle? 5 x 4 = 20 5 x 10 = 50 Fall 2010 14 x5 5 x 10 → 50 5 x 4 → + 20 70 15 5.19: Distributive Property of 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 Fall 2010 16 5.19: Distributive Property of Multiplication Over Addition Use base ten blocks to build a 12 x 23 rectangle. The traditional multidigit multiplication algorithm finds the sum of the areas of two inner rectangles. National Library of Virtual Manipulatives – Rectangle Multiplication Fall 2010 17 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 Fall 2010 18 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. Fall 2010 19 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 Base: 2cm by 3cm; Height: 4cm grouping of the factors does Volume: (2 x 3) x 4 = 24 cm3 not affect the product. National Library of Virtual Manipulatives – Space Blocks Fall 2010 20 Expressions and Equations Fall 2010 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 Fall 2010 What do these tiles represent? 1 unit Area = 1 square unit 1 unit Tile Bin Unknown length, x units Area = x square units 1 unit x units x units Area = x2 square units The red tiles denote negative quantities. Fall 2010 23 Modeling expressions x+5 Tile Bin 5+x Fall 2010 24 Modeling expressions x-1 Fall 2010 Tile Bin 25 Modeling expressions x+2 Tile Bin 2x Fall 2010 26 Modeling expressions x2 + 3x + 2 Fall 2010 Tile Bin 27 Simplifying expressions x2 + x - 2x2 + 2x - 1 Tile Bin zero pair Simplified expression -x2 + 3x - 1 Fall 2010 28 Simplifying expressions 2(2x + 3) Tile Bin Simplified expression 4x + 6 Fall 2010 29 Two methods of illustrating the Distributive Property: Example: 2(2x + 3) Fall 2010 30 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 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. 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. Fall 2010 31 Solving Equations Tile Bin Fall 2010 32 Solving Equations 6.18 The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions. x+3=5 Fall 2010 Tile Bin 33 Solving Equations 6.18 The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions. Pictorial Representation: Symbolic Representation: Condensed Symbolic Representation: x+3=5 x+3=5 ̵3 ̵3 x+3=5 ̵3 ̵3 x=2 x=2 Fall 2010 34 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 Fall 2010 Tile Bin 35 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-step linear equations. 3=x-1 Fall 2010 Tile Bin 36 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-step linear equations. 2x + 3 = 13 Fall 2010 Tile Bin 37 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-step linear equations. Pictorial Representation: Symbolic Representation: Condensed Symbolic Representation: 2x + 3 = 13 2x + 3 = 13 ̵3 ̵3 2x = 10 2 2 2x + 3 = 13 ̵3 ̵3 2x = 10 2 2 x=5 x=5 Fall 2010 38 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-step linear equations. 0 = 4 – 2x Fall 2010 Tile Bin 39 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-step linear equations. Pictorial Representation: Symbolic Representation: Condensed Symbolic Representation: 0 = 4 – 2x 0 = 4 – 2x ̵4 ̵4 -4 = -2x 2 2 0 = 4 – 2x ̵4 ̵4 -4 = -2x -2 -2 2=x -2 = -x 2=x Fall 2010 40 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 Fall 2010 Tile Bin 41 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: 3x + 5 – x = 11 2x + 5 = 11 2x + 5 = 11 -5 -5 2x = 6 2 2 3x + 5 – x = 11 2x + 5 = 11 -5 -5 2x = 6 2 2 x=3 x=3 Fall 2010 42 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) Fall 2010 Tile Bin 43 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: x + 2 = 2(2x + 1) x + 2 = 4x + 2 x + 2 = 4x + 2 -x -x x + 2 = 2(2x + 1) 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 Fall 2010 Condensed Symbolic Representation: 0=x 44 Modeling Multiplication and Division of Fractions Fall 2010 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. Fall 2010 46 Thinking About Multiplication The expression… We read it… It means… It looks like… 23 1 2 3 1 1 2 3 Fall 2010 47 Thinking About Multiplication The expression… 23 1 2 3 1 1 2 3 Fall 2010 We read it… It means… 2 times 3 two groups of three 2 times 1 3 1 times 1 2 3 It looks like… two groups of one-third one-half group of one-third 48 Making sense of multiplication of fractions using paper folding and area models Enhanced Scope and Sequence, 2004, pages 22 - 24 Fall 2010 49 Making sense of multiplication of fractions using paper folding and area models Enhanced Scope and Sequence, 2004, pages 22 - 24 Fall 2010 50 Making sense of multiplication of fractions using paper folding and area models Enhanced Scope and Sequence, 2004, pages 22 - 24 Fall 2010 51 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 Fall 2010 52 Contexts for Modeling Multiplication of Fractions 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? Fall 2010 53 1 1 1 3 2 6 One-third of one-half of a pizza is equal to one-sixth of a pizza. Which meaning of multiplication does this model fit? Fall 2010 54 Another Context for Multiplication of Fractions 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 1 of her half of the 3 race. What portion of the race had Andrea run when she stopped for water? Fall 2010 55 1 1 1 3 2 6 Students need experiences with problems that lend themselves to a linear model. Fall 2010 56 Another Context for Multiplication of Fractions Mrs. Jones has 24 gold stickers that she bought to put on perfect test 1 papers. She took 2 of the stickers out of the package, and then she 1 used 3 of that half on the papers. What fraction of the 24 stickers did she use on the perfect test papers? Fall 2010 57 1 1 1 3 2 6 1 One-third of one-half of the 24 stickers is 6 of the 24 stickers. What meaning(s) of multiplication does this model fit? Problems involving discrete items may be represented with set models. Fall 2010 58 What’s the relationship between multiplying and dividing? 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: 1 62 6 2 Fall 2010 59 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 Children’s Mathematical Power, LEA Publishing, 1998. Fall 2010 60 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. Fall 2010 61 “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. Fall 2010 62 “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 63 Thinking About Division The expression… We read it… It means… It looks like… 20 ÷ 5 1 20 2 Fall 2010 64 Thinking About Division The expression… 20 ÷ 5 We read it… 20 divided by 5 It means… It looks like… 20 divided into groups of 5; 20 divided into 5 equal groups… How many 5’s are in 20? 1 20 2 20 divided by 1 2 20 divided into groups of 1 … 2 How many 1 ’s are 2 in 20? 65 Fall 2010 65 Thinking About Division The expression… 1 1 2 3 We read it… one-half divided by one-third It means… It looks like… 1 2 divided into groups of 1 … 3 ? How many 1 ’s are 3 1 in 2 ? Is the quotient more than one or less than one? How do you know? Fall 2010 66 Contexts for Division of Fractions The Andersons had half of a pizza left after 1 dinner. Their son’s typical serving size is 3 pizza. How many of these servings will he eat if he finishes the pizza? Fall 2010 67 1 1 1 1 2 3 2 1 1 1 2 pizza divided into 3 pizza servings = 1 2 servings 1 serving 1 serving 2 Fall 2010 68 Another Context for Division of Fractions Marcy is baking brownies. Her recipe calls for 1 3 cup cocoa for each batch of brownies. Once she gets started, Marcy realizes she 1 only has 2 cup cocoa. If Marcy uses all of the cocoa, how many batches of brownies can she bake? Fall 2010 69 1 1 1 1 2 3 2 1 cup Three batches (or Two batches (or 1 cup 2 3 3 cup) 2 3 cup) 1 12 batches One batch (or 13 cup) 0 cups Fall 2010 70 Another Context for Division of Fractions 1 2 Mrs. Smith had of a sheet cake left over after her party. She decides to divide the rest of the 1 cake into portions that equal 3 of the original cake. 1 3 How many cake portions can Mrs. Smith make from her left-over cake? Fall 2010 71 What could it look like? 1 1 2 3 Fall 2010 72 What does it look like numerically? Fall 2010 73 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 Fall 2010 74 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. Fall 2010 75 What about the traditional algorithm? Build understanding: 1 Think about 20 ÷ 2 . 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. Fall 2010 76 What about the traditional algorithm? Later, think about divisors with numerators > 1. Think about 1 ÷ 2 . 3 2 How many times could we take 3 from 1? 1 We can take it out once, and we’d have 3 left. We 2 3 could only take half of another from the remaining 3 portion. That’s a total of 2 . 3 2 In each unit, there are sets of . 2 3 Fall 2010 77 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. Fall 2010 78 Using multiple representations to express understanding Given problem Contextual situation Check your solution Solve numerically Fall 2010 Solve graphically 79 Using multiple representations to express understanding of division of fractions Fall 2010 80 Mean: Fair Share and Balance Point Fall 2010 81 Mean: Fair Share 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.” Fall 2010 82 Understanding the Mean 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. Fall 2010 83 Understanding the Mean 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? Fall 2010 84 Understanding the Mean 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? Fall 2010 85 Understanding the Mean 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. Fall 2010 86 Understanding the Mean How did we display our data? 2009 3.17c Fall 2010 87 Understanding the Mean 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? Fall 2010 88 Mean: Balance Point 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. 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.” Essential Knowledge & Skills: • Identify and draw a number line that demonstrates the concept of mean as balance point for a set of data. Fall 2010 89 Where is the balance point for this data set? X Fall 2010 X X X X X 90 Where is the balance point for this data set? X X X X Fall 2010 X X 91 Where is the balance point for this data set? X X X Fall 2010 X X X 92 Where is the balance point for this data set? X X Fall 2010 X X X X 93 Where is the balance point for this data set? 3 is the Balance Point X Fall 2010 X X X X X 94 Where is the balance point for this data set? X Fall 2010 X X X X X 95 Where is the balance point for this data set? Move 2 Steps Move 2 Steps Move 2 Steps Move 2 Steps Fall 2010 4 is the Balance Point 96 We can confirm this by calculating: 2 + 2 + 2 + 3 + 3 + 4 + 5 + 7 + 8 = 36 36 ÷ 9 = 4 Fall 2010 The Mean is the Balance Point 97 Where is the balance point for this data set? If we could “zoom in” on the Move 1 Step The Balance Point is between 10 and 11 Move 2 Steps (closer to 10). Fall 2010 space between 10 and 11, we could continue this process to arrive at a decimal value for the balance point. Move 2 Steps Move 1 Step 98 Mean: Balance Point 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. Fall 2010 99 Assessing Higher-Level Thinking 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. Fall 2010 10 Mean: Fair Share & Balance Point “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 Fall 2010 10 Inequalities Fall 2010 102 Inequalities 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 b) 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 Fall 2010 103 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 Fall 2010 104 Equations vs. Inequalities x=2 x>2 How are they alike? How are they different? So, what about x > 2? Fall 2010 105 Equations vs. Inequalities x=2 x>2 x>2 Fall 2010 106 Open or Closed? x > 16 -5 > y m > 12 n < 341 -3 < j and, which way should the ray go? Fall 2010 107 Equations vs. Inequalities x+2=8 x+2<8 How are they alike? How are they different? So, what about x + 2 < 8? Fall 2010 108 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. Fall 2010 109 Equations vs. Inequalities x+ 2 = 8 x+ 2 < 8 x+ 2 < 8 Fall 2010 110 Inequality Match With your tablemates, find as many matches as possible in the set of cards. Fall 2010 111 X >5 X is greater than 5 SAMPLE MATCH Fall 2010 112 Operations with Integers Fall 2010 113 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” Fall 2010 114 Assessing Higher-Level Thinking 7.3a: The student will model addition, subtraction, multiplication, and division of integers. = -1 =1 What operation does this model? Fall 2010 3 + (-7) = -4 115 Assessing Higher-Level Thinking 7.3a: The student will model addition, subtraction, multiplication, and division of integers. =1 = -1 3 • (-4) = -12 What operation does this model? Fall 2010 116 Assessing Higher-Level Thinking 7.3a: The student will model addition, subtraction, multiplication, and division of integers. 5 5+ -(-17) 17 = =-12 -12 What operation does this model? Fall 2010 117 Assessing Higher-Level Thinking 7.3a: The student will model addition, subtraction, multiplication, and division of integers. 3 • (-5) = -15 What operation does this model? Fall 2010 118 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 ½ or 2. 8 in. 3 in. 5 in. Fall 2010 119 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. Fall 2010 120 Exit Slip 1. Aha... 2. Can’t wait to share… 3. HELP! Fall 2010 121