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Content Deepening 7 th Grade Math September 23, 2013 Jeanne Simpson AMSTI Math Specialist

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Name School Classes you teach What are you hoping to learn today? Welcome 2

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He who dares to teach must never cease to learn. John Cotton Dana 4

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acos2010.wikispaces.com Electronic version of handouts Links to web resources

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Instructi on Content InterventionAssessment Collaboration Five Fundamental Areas Required for Successful Implementation of CCSS 6

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Instruction Deep conceptual understanding Collaborative lesson design Standards for Mathematical Practice Content Fewer standards with greater depth Understanding, focus, and coherence Common and high-demand tasks Intervention Common required response to intervention framework response Differentiated, targeted, and intensive response to student needs Student equity, access, and support Assessment PLC teaching-assessing-learning cycle In-class formative assessment processes Common assessment instruments as formative learning opportunities Collaboration How do we teach? 7

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SMP1 - Make sense of problems and persevere in solving them SMP2 - Reason abstractly and quantitatively SMP3 - Construct viable arguments and critique the reasoning of others SMP4 - Model with mathematics SMP5 - Use appropriate tools strategically SMP6 - Attend to precision SMP7 - Look for and make use of structure SMP8 - Look for and express regularity in repeated reasoning Standards for Mathematical Practice 8

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Capture the processes and proficiencies that we want our students to possess Not just the knowledge and skills but how our students use the knowledge and skills Describe habits of mind of the mathematically proficient student Carry across all grade levels, K-12 What Are The Practice Standards? 9

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√ I already do this. ! This sounds exciting! ? I have questions. Standards of Mathematical Practice 10

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Problem solving Demanding tasks Student understanding Discussion of alternative strategies Extensive mathematics discussion Effective questioning Student conjectures Multiple representations High-Leverage Strategies 11

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Instruction Deep conceptual understanding Collaborative lesson design Standards for Mathematical Practice Content Fewer standards with greater depth Understanding, focus, and coherence Common and high-demand tasks Intervention Common required response to intervention framework response Differentiated, targeted, and intensive response to student needs Student equity, access, and support Assessment PLC teaching-assessing-learning cycle In-class formative assessment processes Common assessment instruments as formative learning opportunities Collaboration What are we teaching? 12

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PICS Share your section and record what you hear Describe one connection you notice How could you use this with students 13

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Critical Focus Areas Ratios and Proportional Reasoning Applying to problems Graphing and slope Standards 1-3 Number Systems, Expressions and Equations Standards 4-10 Geometry Scale drawings, constructions, area, surface area, and volume Standards Statistics Drawing inferences about populations based on samples Standards Probability – Standards 21-24

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Recommend Emphases from PARCC Model Content Framework for Mathematics

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7 th Grade Domains 1.Ratios and Proportional Reasoning 2.The Number System 3.Expressions and Equations 4.Geometry 5.Statistics and Probability 16

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Analysis Tool Content Standard Cluster Which Standards in the Cluster Are Familiar? What’s New or Challenging in These Standards? Which Standards in the Cluster Need Unpacking or Emphasizing? How Is This Cluster Connected to the Other 6-8 Domains and Mathematical Practice? Draw informal comparative inferences about two populations. 7.SP3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between centers by expressing it as a multiple of a measure of variability.

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Unpacking the Standards 18

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Unpacking the Standards Step 1: Target a standard Step 2: Chunk the Main Categories Step 3: Identify all standard components Step 4: Identify the Developmental Progression Step 5: Identify Key Vocabulary Step 6: Add Clarifying Information “To increase student achievement by ensuring educators understand specifically what the new standards mean a student must know, understand, and be able to do. (Unpacking) may also be used to facilitate discussion among teachers and curriculum staff and to encourage coherence…(Unpacking), along with on-going professional development is one of many resources used to understand and teach the CCSS.” -North Carolina Dept of Public Instruction 19

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Why are we Unpacking Standards? To understand what the standards are asking students to know, understand, and be able to do To make time for professional discussion about the standards To build upon and use common terminology when discussing the implementation of the standards Unpacking is standards is not a substitute document for the Common Core Standards, it is a record of the conversation of those who are involved in the process of digging into the standards. 20

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Step 1 – Target a Standard What standard(s) do you need to explore further? Find a group of 2-4 teachers who will explore that topic with you. 21

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2.G.3 Partition circles and rectangles into two, three, or four equal shares Partition circles and rectangles into two equal shares, using the word halves, half of 2.G.3 Partition Builds on 1.G.3 Needed for 3.G.2 partition Equal shares rectangleHalves Half of circle Partition a shape into fourths in different ways Pattern Blocks Fraction Bars/Circles Describe Thirds Third of Fourths Fourth of whole Identical whole 2/2 = one whole Partition circles and rectangles into three equal shares, using the word thirds, third of 2.G.3 Partition circles and rectangles into four equal shares, using the word fourths, fourth of 2.G.3 Describe the whole as two halves, three thirds, four fourths 2.G.3 Recognize that equal shares of identical wholes need not have the same shape 2.G.3 Recognize The final product…. 22

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Example 2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. Step 2: Chunk the Main Categories 1.All Standard(s) in the cluster(s) 2.Identify Key Verbs 2.G.3 Partition circles and rectangles into two, three, or four equal shares DescribeRecognize Partition 23

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Step 3: Identify all standard components Components from CCSS: Analyze nouns and verbs What do students need to do? Include bullets, examples, footnotes, etc. Take standard apart according to the verbs to separate skills within the standard What do the students need to know? lt blue 24

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Example 2.G.3 Partition circles and rectangles into two, three, or four equal shares PartitionRecognizeDescribe Partition circles and rectangles into four equal shares, using the word fourths, fourth of 2.G.3 Describe the whole as two halves, three thirds, four fourths 2.G.3 Recognize that equal shares of identical wholes need not have the same shape 2.G.3 Partition circles and rectangles into three equal shares, using the word thirds, third of 2.G.3 Partition circles and rectangles into two equal shares, using the word halves, half of 2.G.3 25

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Step 4: Identify the Developmental Progression Questions to consider when looking at the developmental progression of the standards… How would you utilize these chunks (blue) for scaffolding toward mastery of the entire standard? Where would you start when teaching this standard? What is the chunk that demonstrates the highest level of thinking? 26

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Vertical Alignment Using the progression document(s) from Ohio Department of Education and CCSS Writing Team: Look to the grade level(s) below to see if the standard is introduced. Look to the grade level(s) above to see if the standard is continued. Code each standard on the poster with: builds on introduced needed for or mastered and the grade level to which the standard aligns. 27

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Introduced? Mastered? Needed for? Builds on? 2.G.3 Partition circles and rectangles into two, three, or four equal shares PartitionRecognizeDescribe Partition circles and rectangles into four equal shares, using the word fourths, fourth of 2.G.3 Describe the whole as two halves, three thirds, four fourths 2.G.3 Recognize that equal shares of identical wholes need not have the same shape 2.G.3 Partition circles and rectangles into three equal shares, using the word thirds, third of 2.G.3 Partition circles and rectangles into two equal shares, using the word halves, half of 2.G.3 Builds on 1.G.3 Needed for 3.G.2 28

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Step 5: Identify Key Vocabulary Identify content vocabulary directly from the standard. Identify additional vocabulary students will need to know to meet the standard. green 29

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2.G.3 Partition circles and rectangles into two, three, or four equal shares Partition circles and rectangles into two equal shares, using the word halves, half of 2.G.3 Partition Builds on 1.G.3 Needed for 3.G.2 partition Equal shares rectangleHalves Half of circle Describe Thirds Third of Fourths Fourth of whole Identical whole Partition circles and rectangles into three equal shares, using the word thirds, third of 2.G.3 Partition circles and rectangles into four equal shares, using the word fourths, fourth of 2.G.3 Describe the whole as two halves, three thirds, four fourths 2.G.3 Recognize that equal shares of identical wholes need not have the same shape 2.G.3 Recognize Partition circles and rectangles into two equal shares, using the word halves, half of 2.G.3 30

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Step 6: Add Clarifying Information Kid-friendly language to add clarity Clarifying pictures, words, or phrases Definitions, examples Symbols, formulas, pictures, etc. CAUTION: do not replace important vocabulary that is included in the standard. yellow 31

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2.G.3 Partition circles and rectangles into two, three, or four equal shares Partition circles and rectangles into two equal shares, using the word halves, half of 2.G.3 Partition Builds on 1.G.3 Needed for 3.G.2 partition Equal shares rectangleHalves Half of circle Partition a shape into fourths in different ways Pattern Blocks Fraction Bars/Circles Describe Thirds Third of Fourths Fourth of whole Identical whole 2/2 = one whole Partition circles and rectangles into three equal shares, using the word thirds, third of 2.G.3 Partition circles and rectangles into four equal shares, using the word fourths, fourth of 2.G.3 Describe the whole as two halves, three thirds, four fourths 2.G.3 Recognize that equal shares of identical wholes need not have the same shape 2.G.3 Recognize Partition circles and rectangles into two equal shares, using the word halves, half of 2.G.3 32

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Transfer Unwrapping to Chart 33

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2.G.3 Partition circles and rectangles into two, three, or four equal shares Partition circles and rectangles into two equal shares, using the word halves, half of 2.G.3 Partition Builds on 1.G.3 Needed for 3.G.2 partition Equal shares rectangleHalves Half of circle Partition a shape into fourths in different ways Pattern Blocks Fraction Bars/Circles Describe Thirds Third of Fourths Fourth of whole Identical whole 2/2 = one whole Partition circles and rectangles into three equal shares, using the word thirds, third of 2.G.3 Partition circles and rectangles into four equal shares, using the word fourths, fourth of 2.G.3 Describe the whole as two halves, three thirds, four fourths 2.G.3 Recognize that equal shares of identical wholes need not have the same shape 2.G.3 Recognize Partition circles and rectangles into two equal shares, using the word halves, half of 2.G.3 Main Idea of Standard Key Verbs Take standard apart according to the verbs to separate skills within the standard. Use all components of standard. Put in a logical sequence Vertical alignment Vocabulary Clarifying information, student-friendly 34

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The Number System 35

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The standards stress not only procedural skill but also conceptual understanding, to make sure students are learning and absorbing the critical information they need to succeed at higher levels – rather than the current practices by which many students learn enough to get by on the next test, but forget it shortly thereafter, only to review again the following year. ( CCSSI, 2012 ) 36

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Mathematics consists of pieces that make sense; they are not just independent manipulation/skills to be practiced and memorized – as perceived by many students. These individual pieces progress through different grades (in organized structures we called “flows”) and can/should be unified together into a coherent whole. Jason Zimba, Bill McCallum 37

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Mathematical Fluency 38

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Fluency The word fluent is used in the Standards to mean fast and accurate. Fluency in each grade involves a mixture of just knowing some answers from patterns (e.g., “adding 0 yields the same number”), and knowing some answers from the use of strategies. Progressions for the Common Core State Standards in Mathematics 39

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Fluency Fluent in the standards means “fast and accurate.” It might also help to think of fluency as meaning more or less the same as when someone is said to be fluent in a foreign language. To be fluent is to flow; fluent isn’t halting, stumbling, or reversing oneself. Jason Zimba 40

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Fluency Expectations 41 GradeRequired Fluency KAdd/subtract within 5 1Add/subtract within 10 2Add/subtract within 20 Add/subtract within 100 (pencil and paper) 3Multiply/divide within 100 Add/subtract within Add/subtract within 1,000,000 5Multi-digit multiplication 6Multi-digit division (6.NS.2) Multi-digit decimal operations (6.NS.3) 7Solve px + q = r, p(x + q) = r

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Most of the numbers you have worked with in math class this year have been greater than or equal to zero. However, numbers less than zero can provide important information. Where have you seen numbers less than zero outside of school? Integers

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7NS - Understanding 7.NS.1b – Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. 7.NS.1c – Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). 7.NS.2a – Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. 7.NS.2b – Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. 44

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K–6 Geometry 6-8 Statistics and Probability 6–7 Ratios and Proportional Relationships 6–8 Expressions and Equations 6-8 Number Systems 3-5 Number and Operations: Fractions These are the documents currently available. They are working on documents for the other domains (Functions, Geometry 7-8). Progressions Documents

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Jigsaw Read your assigned section Chart paper ▫Summarize what you read. ▫How can this document help you in your classroom? Be prepared to share 46

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Adding Integers 47

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Opposites 48

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Subtracting Integers 49

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Subtracting Integers 50

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Fractions Difficulty with learning fractions is pervasive and is an obstacle to further progress in mathematics and other domains dependent on mathematics, including algebra. It has also been linked to difficulties in adulthood, such as failure to understand medication regimens. National Mathematics Panel Report,

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Fractions “Students who are asked to practice the algorithm over and over…stop thinking. They sacrifice the relationships in order to treat the numbers simply as digits.” Imm, Fosnot, Uittenbogaard (2012) 52

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Unit Fractions 53

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Fraction Multiplication in Grade 5 54

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Fraction Multiplication in Grade 5 55

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Fraction Multiplication in Grade 5 56

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Fraction Multiplication in Grade 5 57

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5 th Grade Division 58

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5 th Grade Division 59

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5 th Grade Division Problems How much chocolate will each person get if 3 people share ½ pound equally? 60

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5 th Grade Division Problems How many 1/3 cup servings are in 2/3 cups of raisins? 61

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Fraction Division in Grade 6 6.NS.1 – Interpret and compute quotients of fractions, and solve word problems involving division of fractions, e.g., by using visual fraction models and equations to represent the problem. Examples: ▫Create a story context… ▫Use a visual fraction model to show the quotient… ▫Explain division using its relationship with multiplication ▫Sample problems 62

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6 th Grade Division 63

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6 th Grade Division 64

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Ratios and Proportional Relationships 65

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Kim and Bob ran equally fast around a track. Kim started first. When she had run 9 laps, Bob had run 3 laps. When Bob had run 15 laps, how many laps had Kim run? Explain your reasoning. 66

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Solving Proportions SolveKanold, p. 94 The traditional method of creating and solving proportions by using cross- multiplication is de- emphasized (in fact it is not mentioned in the CCSS) because it obscures the proportional relationship between quantities in a given problem situation. If two pounds of beans cost $5, how much will 15 pounds of beans cost? 67

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Ratios and Proportional Relationships Progression, pages 6-7 Although it is traditional to move students quickly to solving proportions by setting up an equation, the Standards do not require this method in Grade 6. There are a number of strategies for solving problems that involve ratios. As students become familiar with relationships among equivalent ratios, their strategies become increasingly abbreviated and efficient. 68

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6.RP.3 Use ratio and rate reasoning to solve real- world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. a.Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. b.Solve unit rate problems including those involving unit pricing and constant speed. c.Find a percent of a quantity as a rate per 100; solve problems involving finding the whole, given a part and the percent. d.Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. 69

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1.The ratio of free throws that Omar made to the ones he missed at practice yesterday was 7:3. If he attempted 90 free throw at practice, how many free throws did Omar make? 70 made 7 missed 3 Attempted 90 9 x 7 = ÷ 10 = 9 Omar made 63 free throws

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2.At FDR High School, the ratio of seniors who attend college to those who do not is 5:2. If 98 seniors do not attend college, how many do? 71

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3.At Mesa Park High School, the ratio of students who have driver’s licenses to those who don’t is 8:3. If 144 students have driver’s licenses, how many students are enrolled at Mesa Park High School? 72

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4.Of the black and blue pens that Mrs. White has in a drawer in her desk, 18 are black. The ratio of black pens to blue pens is 2:3. When Mrs. White removes 3 blue pens, what is the new ratio of black pens to blue pens? 73

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Expressions and Equations 74

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Understanding 7.EE.2 – Understand that rewriting an expression in different forms in a problem context can shed light on the problem, and how the quantities in it are related. 75

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Fluency Expectations 76 GradeRequired Fluency KAdd/subtract within 5 1Add/subtract within 10 2Add/subtract within 20 Add/subtract within 100 (pencil and paper) 3Multiply/divide within 100 Add/subtract within Add/subtract within 1,000,000 5Multi-digit multiplication 6Multi-digit division (6.NS.2) Multi-digit decimal operations (6.NS.3) 7Solve px + q = r, p(x + q) = r

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Fluency Expectations 77 GradeRequired Fluency KAdd/subtract within 5 1Add/subtract within 10 2Add/subtract within 20 Add/subtract within 100 (pencil and paper) 3Multiply/divide within 100 Add/subtract within Add/subtract within 1,000,000 5Multi-digit multiplication 6Multi-digit division (6.NS.2) Multi-digit decimal operations (6.NS.3) 7Solve px + q = r, p(x + q) = r

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The Mystery Bags Game Solving Equations

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The Mystery Bags Game There once was a king who loved to watch his many bags of gold.

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But it can get very boring watching gold all day, so he had the court jester make up games for him to pass the time. The game the king loves best is the Mystery Bags game.

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First, the jester takes one or more empty bags and fills each bag with the same amount of gold. These bags are called the “mystery bags.”

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Next, the jester digs into his collection of lead weights. He takes out his pan balance and places some combination of mystery bags and lead weights on the two pans so that the sides balance. The game is to figure out the weight of each mystery bag.

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Your Task The game sound rather easy, but it can get very difficult for the king. See if you can win the mystery bags game in the various situations described on your worksheet by figuring out how much gold there is in each mystery bag. Explain how you know you are correct. You may want to draw diagrams to show what’s going on.

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Write an equation to represent this situation Use your methods from Problem 3.2 to find the number of gold coins in each pouch Write down a similar method using the equation that represents this situation

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Question 1 There are 3 mystery bags on one side of the balance and 51 ounces of lead weights on the other side. 51 oz.

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2 There are 1 mystery bag and 42 ounces of weights on one side, and 100 ounces of weights on the other side.

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3 There are 8 mystery bags and 10 ounces of weights on one side, and 90 ounces of weights on the other side.

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4 There are 3 mystery bags and 29 ounces of weights on one side, and 4 mystery bags on the other side.

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5 There are 11 mystery bags and 65 ounces of weights on one side, and 4 mystery bags and 100 ounces of weights on the other side.

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6 There are 6 mystery bags and 13 ounces of weights on one side, and 6 mystery bags and 14 ounces of weights on the other side. (The jester could get in a lot of trouble for this one!)

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7 There are 15 mystery bags and 7 ounces of weights on both sides. (At first the king thought this one was easy, but then he found it to be incredibly hard.)

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8 The king wants to be able to win easily all of the time, without calling you in. Therefore, your final task in this assignment is to describe in words a procedure by which the king can find out how much is in a mystery bag in any situation.

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7.EE.4 – Use variables to represent quantities in real-world or mathematical problems, and construct simple equations and inequalities to solve problems by reasoning about the quantities. 106

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Mathematics Assessment Project Tools for formative and summative assessment that make knowledge and reasoning visible, and help teachers to guide students in how to improve, and monitor their progress. These tools comprise: Classroom Challenges: lessons for formative assessment, some focused on developing math concepts, others on non-routine problem solving. Professional Development Modules: to help teachers with the new pedagogical challenges that formative assessment presents. Summative Assessment Task Collection: to illustrate the range of performance goals required by CCSSM. Prototype Summative Tests: designed to help teachers and students monitor their progress, these tests provide a model for examinations that may replace or complement current US tests /

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Steps to Solving EquationsProjector Resources Writing Algebraic Expressions P-108 Area of rectangle = _ _ _ _ _ _ _ _ _ _ _ _

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Steps to Solving EquationsProjector Resources Writing Algebraic Expressions P-109 Perimeter of rectangle = _ _ _ _ _ _ _ _ _ _ _ _

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Steps to Solving EquationsProjector Resources Writing Algebraic Expressions P-110 Which two expressions are equivalent?

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Steps to Solving EquationsProjector Resources Which Equations Describe The Story? P-111 A pencil costs $2 less than a notebook. A pen costs 3 times as much as a pencil. The pen costs $9 Which of the four equations opposite describe this story? Let x represent the cost of notebook.

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Geometry 112

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The van Hiele Theory of Geometric Thought 113

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Statistics and Probability Resources 115

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Understanding 7.SP.1 – Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. 116

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After surveying 83 students in 3 classes, 70% responded that girls should be allowed to go to lunch two minutes early every day and boys will go at the regular time. Do you think this is an accurate statistic? Who do you think the sample population was?

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Each group will need to assign the following roles: Facilitator – keeps group on task and ensures equal participation Materials Manager – collects and returns materials Recorder – writes group answer on chart paper Reporter – presents group answer to the class

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Discuss and complete the handout as a group. Begin with the multiple choice questions. Choose one biased survey to present to the class on chart paper. Include the following in your presentation: Original survey Why you think it is biased How you would correct it

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Question Population Sample group

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In a poll of Mrs. Simpson’s math class, 67% of the students say that math is their favorite academic subject. The editor of the school paper is in the class, and he wants to write an article for the paper saying that math is the most popular subject at the school. Explain why this is not a valid conclusion, and suggest a way to gather better data to determine what subject is most popular.

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Illustrative Mathematics 125 Illustrative Mathematics provides guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students experience in a faithful implementation of the Common Core State Standards, and by publishing other tools that support implementation of the standards.

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What Do You Expect? Compound Probability

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How is probability used in real life?

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Event Theoretical probability Experimental probability Outcome

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What’s in the Bucket?

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Without looking in or emptying the bucket, how could we determine the fraction of blocks that are red, yellow, or blue?

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Problem 1.1 How many blocks drawn by your class were blue? How many were yellow? How many were red? Which color block do you think there are the greatest of in the bucket? Which color block do you think there are the least number of?

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Problem 1.1 Based on our experimental data, predict the fraction of blocks in the bucket that are blue, that are yellow, and that are red.

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Problem 1.1 How do the fraction of blocks that are blue, yellow, and red compare to the fractions of blue, yellow, and red drawn during the experiment?

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Match / No-Match Rules Spin the spinner twice for each turn. If both spins land on the same color, you have made a MATCH. Player A scores 1 point. If the two spins land on different colors, you have made a NO-MATCH. Player B scores 2 points.

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Experimental Probability of Match number of turns that are matches total number of turns

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Experimental Probability of No-Match number of turns that are no-matches total number of turns

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What are the possible outcomes in this game? Color on 1 st spin – Color on 2 nd spin Are all outcomes equally likely?

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Theoretical Probability of No-Match number of outcomes that are no- matches total number of turns

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Theoretical Probability of Match number of outcomes that are matches total number of turns

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A. Compare the experimental and theoretical probabilities for match and for no-match. B. Is Match/No-Match a fair game ? If you think the game is fair, explain why. If you think it is not fair, explain how the rules could be changed to make it fair.

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Making Purple BROWN YELLOW RED ORANGE BLUEGREEN

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Jeanne Simpson UAHuntsville AMSTI Contact Information 144

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Feedback Praise Question Polish 145

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