Putting the Mathematical Practices Into Action
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  • Survey participants: first timers, math coaches, classroom teachers, central office, principals, etc… This will allow us to modify presentation accordingly.
  • Thumbs up if you agree with these norms. Are there other norms we need to add so that we have the best possible learning experience for all?
  • We will not use all slides but we have included additional slides for your convenience so that you may modify accordingly to your needs for a school or district. Example: If used in PLC’s – may want to take one Practice Standard at a time and develop it more thoroughly.
  • Share outcomes
  • Discuss
  • Discuss
  • Increase attention on problem solving, discussion, and justification of thinking and decrease attention on rote practice, rote memorization of rules, and teaching by telling
  • Hand out books and posters
  • Discuss
  • Practices are grouped into sub groups: reasoning & explaining; modeling and using tools; seeing structure and generalizing Gray represent overarching habits of mind of a productive Mathematical Thinker © 2012 Karen A. Blase and Dean L. Fixsen
  • Think about what this standard means and looks like in a classroom. Here are some questions to help you with your reflection.
  • These are reflective questions notes in the book. Participants turn and talk.
  • Have participants brainstorm strategies student use to solve problems. Then click and the list noted in book will appear.
  • In this question, discussion occurs on how to find an answer, not necessarily the answer. This is followed up over the next few days with different questions.
  • Using the same data, different questions can be asked that lead to strong classroom discussion.
  • Quick responses do not require mathematical thinking, do not create proficient mathematicians and do not promote the practices.
  • What would be the benefit of doing this activity with teachers?
  • Participant refer to book for additional ideas to develop the practice.
  • Turn and Talk
  • Think about what this standard means and looks like in a classroom. Here are some questions to help you with your reflection.
  • Have participants create a number web. Carousel of webs: think about fractions, decimals, whole numbers…choose a number and then as a group at least two ways to represent it before the buzzer. Rotate to the next poster Have groups take post it notes with them. If they disagree with a representation, they can note it and tell why.
  • This is an example from the book on pg. 34.
  • What misconceptions does this student have? The calculation is correct, but the student does not have an understanding of multiplication.
  • Student created a problem that shows an understanding of multiplication.
  • Writing problems to contextualize math helps students with their ability to solve problems because they strengthen their understanding of the link between real situations and math equations.
  • Each student is given a pinch card, which is created with an index card that is printed with the operation signs in the same location on the front and back of the card. Math word problems are posed to students who then pinch the sign they would use to solve the problem. Word problems should match students’ grade levels and math computation skills.
  • Often the key word or phrase in a problem suggests an operation that is incorrect. For example: Maxine took the 28 stickers she no longer wanted and gave them to Zandra. Now, Maxine has 37 stickers left . How many stickers did Maxine have to begin with? Especially when you get away from overly simple problems found in primary textbooks and a child that has been taught to rely on key words is left with no strategy. The most important approach to solving any contextual problem is to analyze the structure and make sense of it. The key word approach encourages students to ignore the meaning of the problem and look for an easy way out.
  • To successfully interpret and solve math problems, students must be able to decontextualize and contextualize problems.
  • Labeling answers: Numeric part of the answer attests to student’s computational accuracy Label attests to whether the student knows what the label represents
  • Participant refer to book for additional ideas to develop the practice.
  • Think about what this standard means and looks like in a classroom. Here are some questions to help you with your reflection.
  • Discuss
  • Three strategies for constructing and critiquing arguments.
  • Students are presented with four math concepts and asked to decide on the one that should be eliminated based on mathematical fact or reason. There may be more than one way to eliminate an item. The challenge s to construct a clear argument for eliminating the item selected. Do these, share, create their own.
  • Teacher poses a math statement and asks students to agree or disagree with the statement. Students must include math data or reasoning to support their decision. This student agrees that 75% is more than 2/3, converting 75% to a fraction and then drawing a diagram to compare fractions.
  • This student disagrees that 5 nickels are worth more than 3 dimes. She provides information about the value of the coins and clearly explains her computations to prove her thinking.
  • Cards for each table. Each participant takes a card and explains if they agree or disagree and why.
  • Students are presented with sample arguments with flawed or incomplete logic. Students work individually or with partners to improve the sample. This student critiques the argument and offers some ideas as to why it is faulty. This helps avoid the “teacher is always right” mentality.
  • The act of constructing arguments challenges students to think about the math they are doing and leads them to discover mistakes as well as insights while they struggle with the construction of their arguments.
  • Participant refer to book for additional ideas to develop the practice.
  • Think about what this standard means and looks like in a classroom. Here are some questions to help you with your reflection.
  • Students need lots of experiences constructing math models. Students are challenged to think about the math and determine a way to represent it.
  • How would you model this situation?
  • Result unknown 3 x 6 = ?
  • number in group unknown 18 divided by 3 = 6
  • Number of boxes unknown 18 divided by 6 = 3
  • Participant refer to book for additional ideas to develop the practice.
  • Formal and informal Spontaneous , informal interview can yield insights about the reasoning and skills. Formal interviews might be done at a specific time which students bring their model to our desk and we pose questions like the following:
  • Think about what this standard means and looks like in a classroom. Here are some questions to help you with your reflection. If we attempt to place each of these standards in a separate compartment, we will surely become frustrated and confused.
  • While we may immediately think of tools as math devices like rulers, compasses, protractors etc…, the CCSS Standards for Mathematical Practices have a much broader definition of tools. Tools are what support students to perform the task. Concrete materials like base-ten blocks, connecting cubes, and 10 x 10 grids can be tools, so grid paper, number lines, and hundreds charts. Calculators, pencil and paper, and even mental math are also tools. Tools enhance our students’ mathematically power by assisting them as they perform tasks.
  • Students begin to decide which tools will best meet their needs.
  • Students benefit from opportunity to select a tool that makes sense for the math task and to evaluate which tool is most efficient for that task. Not only do our students need to be able to select appropriate tools, they must be able to effectively use those tools. – page 80-81 – using tools appropriately!
  • Number lines are powerful tools for computation and estimation tasks. The ability to view th quantities on a number line proves the comparison. It’s close too… helps students justify their decision, it’s visually exploring numbers.
  • Students need a clear understanding of the tools purpose. A common error occurs when our students do not accurately align the beginning of the ruler with the beginning of the object.
  • The understanding of equivalent fractions enhances our students abilities to use rulers accurately.
  • Increase or decrease the complexity. Simplify task, students could identify just one set of number partners or put fewer numbers in the box. To increase complexity, include numbers that don’t have partners or ask students to create their own number sets and target numbers.
  • Develops computational fluency and mental math skills. 734 x 82 – paper and pencil 63 x 4 - 1/4 + 2/8 930 ÷ 3
  • Participant refer to book for additional ideas to develop the practice.
  • Think about what this standard means and looks like in a classroom. Here are some questions to help you with your reflection.
  • Math relies on precision in computation and communication. To communicate effectively about math content, students need to know the words that express that content. Students must understand the math and know that words that express that understanding. Understanding symbols is more than the ability to read them. Being able to name a symbol does not ensure students understand the meaning of the symbol.
  • When is precision important? Pose situations in which students must decide whether an estimate or an exact answer is needed. Students develop arguments for whether an exact or an estimated answer makes more sense in the situation
  • Partner talk
  • Quick and effective way to explore math ideas and expand math vocabulary. Questioning: Why did you think of that word? How does it relate to our word of the day? What is…? Can you give me an example of…? When have we talked about … before?
  • Students identify similarities and differences between math concepts. Sort and label cards
  • Give each pair of students a set of word cards and have partners take turns picking a card at random. Have students describe the word, without actually saying the word. The partner should figure out the mystery word from the clues. Continue to give clues until the word is identified. Talk about how to differentiate.
  • Place math expressions, equations, or inequalities on index cards Students select a card and explain it to their partner or explain in writing
  • Word boxes work well for assessment and instruction Table group activity
  • Participant refer to book for additional ideas to develop the practice.
  • Think about what this standard means and looks like in a classroom. Here are some questions to help you with your reflection.
  • There is structure in math. People who see that structure find that math makes sense. Proficient students: See that numbers are flexible Understand properties Recognize patterns and functions
  • See solutions on pages111-112
  • Students deepen their understanding of fractions as they decompose 1 ½
  • Discussion on how this can be used in the classroom.
  • Ratio tables make sense when two pieces of data are connected. Supports students as they develop understanding and automaticity with multiplication and division.
  • A number line help students visualize properties.
  • Participant refer to book for additional ideas to develop the practice.
  • Think about what this standard means and looks like in a classroom. Here are some questions to help you with your reflection.
  • Students find ways to minimize their efforts in mathematics through shortcuts after discovering repetition.
  • What conclusion can your draw? Page 121 Check your prediction with your partner! When adding by one it is always the next number in the counting squence.
  • Share objectives
  • Memorize math rules without understanding. These are tricks to make math earsier but there are no “tricks” in math. It is the understanding of math that makes it easier. Page 124 - more
  • What do you notice? Do you see any pattern? Did you notice anything interesting about the solution? It is the same number just with a 1 in front of it. The one represents a ten.
  • Organize data into table so it is easier to see patterns – ultimately to discover insights. Pose problems in which students are challenges to extend patterns and look for a generalization, or algebraic relationship, to explain the pattern.
  • It is important for students to discover math rules, generalization, and shortcuts. The goal is to set up investigations that allow them ti have “ah-ah” moments. This slide contains some critical components of these investigations.
  • Participant refer to book for additional ideas to develop the practice.
  • Share objectives
  • Additional task to practice using Standards for Mathematical Practice – if time permits.
  • Additional task to practice using Standards for Mathematical Practice – if time permits.
  • What questions do you have?

Putting the Mathematical Practices Into Action Presentation Transcript

  • 1. Fall 2013 Putting the Mathematical Practices Into Action
  • 2. Welcome “Who’s in the Room”
  • 3. Norms • Listen as an Ally • Value Differences • Maintain Professionalism • Participate Actively
  • 4. http://maccss.ncdpi.wikispaces.net/Home Materials Found on Mathematics Wiki
  • 5. Session Outcome • Understand the Standards for Mathematical Practice • Explore strategies for implementing the Standards effectively
  • 6. What does it mean to be Mathematically Proficient?
  • 7. Turn and Talk • Are students who can remember formulas or memorize algorithms truly mathematically proficient, or are there other skills that are necessary? • Is the correct answer the ultimate goal of mathematics, or do we expect a greater level of competence?
  • 8. For the first time, mathematical processes are elevated to essential expectations, changing our view of math to encompass more than just content. The goal now is to apply, communicate, make connections, and reason about math content rather than simply compute.
  • 9. What are The Standards for Mathematical Practice?
  • 10. What has been your biggest challenge with the implementation of The Standards for Mathematical Practice?
  • 11. Overarching habits of mind of a productive Mathematical Thinker
  • 12. Exploring Standard 1 Make Sense of Problems and Persevere in Solving them.
  • 13. Understanding the Standard • What do we do each day in our classroom to build mathematical thinkers? • What do we do to keep our students actively engaged in solving problems? • How do we help our students develop positive attitudes and demonstrate perseverance during problem solving?
  • 14. How Do We Get There? Brainstorming Strategies… 1. Choose an Operation 2. Draw a Picture 3. Find a Pattern 4. Make a Table 5. Guess and Check 6. Make an Organized List 7. Use Logical Reasoning 8. Work Backward Page 11
  • 15. The Holiday Tree The Partin family counted the different types of ornaments on the town’s holiday tree. Here is the list of what they saw. stars – 24 gingerbread men – 14 snowflakes – 12 reindeer – 18 candy canes – 6 Six of the reindeer had red noses. What fraction of the reindeer had red noses? Tell how you would get the answer. Page 14
  • 16. The Holiday Tree The Partin family counted the different types of ornaments on the town’s holiday tree. Here is the list of what they saw. stars – 24 gingerbread men – 14 snowflakes – 12 reindeer – 18 candy canes – 6 What fraction of the ornaments were snowflakes? What fraction of the ornaments were edible? If 6 of the stars were silver, what fractions of the stars were not silver? Page 14
  • 17. What questions could you ask? Look on page 19 for some additional suggestions.
  • 18. What questions could you ask? Shipley Aquarium Admission Cost Adults - $8.00 Children (ages 3 and over) - $6.50 Children (ages 2 and under) – Free Look on page 19 for some additional suggestions.
  • 19. Traditional Problems vs. Rich Problems • We can ask questions that stifle learning by prompting a quick number response. – What is the answer to number 3 on your worksheet? – What is 5 x 4? • We can ask questions that promote discussion, thinking, and perseverance.
  • 20. Sort the math questions. Check your arrangement on page 22
  • 21. page 26
  • 22. Reflecting on strengthening student problem solving experience… 1. Do I routinely provide opportunity for my students to share their solutions and processes with partners, groups, and the whole class? 2. Do I show my students that I value process (how they did it) rather than simply the correct answer? 3. Do I pose problems that require perseverance? Do I use thoughtful questions to guide and encourage students as they struggle with problems?
  • 23. Exploring Standard 2 Reason abstractly and quantitatively.
  • 24. Understanding the Standard • What can we do in our classrooms each day to help students build a strong understanding of numbers (quantities)? • How do we help students convert problems to abstract representations? • What can we do to help students understand what numbers stand for in a given situation?
  • 25. How Do We Get There? • Number Webs • Headline Stories – Post It – Book It – Reverse It – Match It – Question It • Pinch Cards
  • 26. Let’s Create a Number Web! Select a number.
  • 27. Number Web Number Webs encourage flexibility with numbers. page 34
  • 28. Headline Stories • Headlines sum up a story. • Equations are like newspaper headlines—short and to the point. • Equations are connected to word problems the same way a headline is connected to a news story. • Using headlines can help you see students understandings and misunderstandings.
  • 29. Headline Stories page 36
  • 30. Headline Stories page 36
  • 31. Headline Stories Headline: 52÷4 = Headline stories can be as easy or as difficult as you make them! Students might be asked to write problems about equations that include fractions, decimals, percents, or variables. Let’s look at some variations found on page 39.
  • 32. Pinch Cards Pinch cards are an all-pupil response technique. There were 6 soccer teams in the league and 12 players on each team. How many players were in the league? The 4 members of the High Rollers Bowling Team scored 120, 136, 128, and 162. What was the team’s mean score? page 41
  • 33. Avoiding Key Words • Key words are misleading. • Many problems have no key words. • The key word strategy sends a terribly wrong message about doing mathematics. A sense making strategy will always work. Van de Walle & Lovin (2006)
  • 34. Read and Discuss page 33 • What is contextualization and decontextualization? • Why is it important? • Discuss at your table.
  • 35. Contextualize and Decontextualize 120 students and 5 chaperones went on the field trip. Each bus held 35 people. How many buses were needed? Decontextualize: consider the data, the action of the problem, and create an equation to represent the problem in an abstract way Contextualize: refer back to the context of the problem to determine if the answer makes sense
  • 36. • Understanding the units and quantities within a problem is an important factor in making sense of the numbers within the problem. • Labeling answers forces students to refer back to the context of the problem.
  • 37. page 40
  • 38. Exploring Standard 3 Construct viable arguments and critique the reasoning of others.
  • 39. Understanding the Standard • What do we do in the classroom to get students to justify their answer and defend their process for finding the answer? • How do we help students understand math skills and concepts so they can construct viable arguments? • How do we help students consider and judge the reasonableness of other answers and strategies?
  • 40. How Do We Get There? • Eliminate It • Agree or Disagree? • My 2 Cents
  • 41. Eliminate It! • As a group, decide on the concept that should be eliminated with reasoning or math data to back up your decision. • There may be more than one way to eliminate an item! • Create your own.
  • 42. Eliminate It page 50
  • 43. Agree or Disagree? • 75% is more than 2/3. • Tell why you agree or disagree.
  • 44. Agree or Disagree?
  • 45. Agree or Disagree?
  • 46. Agree or Disagree? • Jim has 12 pencils and Annie has 8. Jim has more than Annie. • 7 + 3 and 4 + 6 are the only ways to make 10. • 9 is an even number. • 6 tens and 3 ones is the same as 5 tens and 13 ones. • 3 jars of peanut butter for $7.50 is a better deal than 4 jars of peanut butter for $10.20. page 53
  • 47. My 2 Cents
  • 48. Constructing Arguments • Read page 44-46 • What is the difference between an assertion and an argument? • Be prepared to share your thinking.
  • 49. Assertion vs. Argument • Assertion: a statement of what students want us to believe without support or reasoning. – The answer is correct “because it is,” “because I know it,” or “because I followed the steps.” • Argument: a statement that is backed up with facts, data, or mathematical reasons • Constructing viable arguments is not possible for students who lack an understanding of math skills and concepts.
  • 50. Page 58
  • 51. Exploring Standard 4 Model with mathematics.
  • 52. Understanding the Standard • As teachers, we model with mathematics routinely in our classrooms. Should students be able to model? Why? • How do students modeling mathematics look? • How does student modeling of mathematics affect instruction?
  • 53. How Do We Get There? • Model It • Part-Part-Whole mats (addition & subtraction) • Bar Diagrams (multiplication & division) • Bar Diagrams (solving equations)
  • 54. How would you model… • 123 + 57 • 1 – 1 3 • 3.4 + 5.07
  • 55. Part-Part-Whole Mat • There were 2 yellow lollipops and 3 red lollipops. How many lollipops were there?
  • 56. Part-Part-Whole Mat • There were 6 children. 3 were boys. How many were girls?
  • 57. Part-Part-Whole Mat • There were 5 cupcakes. Jan ate some. There were 2 left. How many did she eat?
  • 58. Bar Diagram • There are 3 boxes with 6 toys in each box. How many toys are there? 6 6 6
  • 59. Bar Diagram • 18 toys are packed equally into 3 boxes. How many toys are in each box? 18
  • 60. Bar Diagram • 18 toys are packed 6 to a box. How many boxes are needed? 18
  • 61. Bar Diagram • 18 toys are packed 6 to a box. How many boxes are needed? 18 6 6 6
  • 62. Page 74
  • 63. Assessment Tips • Tell me what your model represents. • Why did you choose this model? • Did creating a model help you any way? If so, how? • Did you get any insights by looking at your model? • Is there another way you might model this problem or idea? How? Page 75
  • 64. Exploring Standard 5 Use appropriate tools strategically.
  • 65. Understanding the Standard • What are tools used by our students? • Why is it important to use tools? • Tools enhance our students’ mathematical power by assisting them as they perform tasks. • The ability to select appropriate tools is an important reasoning skill.
  • 66. Which tool is more efficient? There is often more than one tool that will work for a task, but some tools are more efficient than others. Paper & Pencil Mental Math Calculator
  • 67. Solve using your assigned tool! 1. 5 x 6 2. 23 x 15 3. Estimate the cost of 2 pies @ $3.75 each Cereal @ $3.20 each Milk @ $1.79 gal Bananas @ 59 cents/lb 1. 236 x 0 x 341 2. What comes next 3, 7, 15, 31, ___ 3. A local TV store had a sale on TV’s. They sold 7 for $1,699.95 each. They made a profit of $169.00 on each TV. What did the store pay for the 7 TVs? A. $1,183.00 C. $13,082.65 B. $10,716.65 D. $11,899.65
  • 68. How Do We Get There? • Use tools appropriately • Number Lines (It’s Close to…) • Rulers (broken ruler, magnified inch) • Mental Math – Number Partners – In My Head?
  • 69. • Students benefit from opportunity to select a tool that makes sense for the math task and to evaluate which tool is most efficient for that task. • Not only do our students need to be able to select appropriate tools, they must be able to effectively use those tools. (page 80-81)
  • 70. Number Lines
  • 71. Rulers Page 84
  • 72. Folding Paper • Fold the strip in half. Open it, and mark ½ at the center fold. • Refold the strip in half and fold it in half again. Label 1/4, 2/4, 3/4 on the three folds.
  • 73. Questions • Are the sections equal in size? • Do the fraction labels make sense? Why? • Where is 0? Why? • Where is 1? Why? • Why are 1/2 and 2/4 on the same fold?
  • 74. Folding Paper • Refold the paper and then fold it in half one more time. • Open the paper, place a mark on each fold and indicate what each of the new marks represent.
  • 75. Folding Paper • Why is there more than one fraction on some folds? • Does it make sense that those fractions are on the same fold? Why? • Which of those fractions is easiest to understand? Would you say 1/2 or 2/4 or 4/8? Why?
  • 76. Number Partners (Mental Math) • Find a Number Partner that Makes 10 5 4 9 3 6 1 7 2 5 8 Page 86 What are some modifications for this task?
  • 77. In My Head? (Mental Math) Do I use paper & pencil or do it in my head? –734 x 82 –63 x 4 –1/4 + 2/8 –930 ÷ 3 Page 86-87 Students need to identify tools that increase their efficiency with math tasks.
  • 78. Page 88-89
  • 79. Exploring Standard 6 Attend to precision.
  • 80. Understanding the Standard • Why is precision important in mathematics? • What does it mean to be precise? • What can we do in the classroom to promote precise communication in mathematics?
  • 81. How Do We Get There? • Estimate and Exact • Vocabulary – Word Webs – Word Walls – Sort and Label – Mystery Words – Translate the Symbol – Word Boxes
  • 82. Estimate and Exact • Buying bags of candy to put in party treat bags • Measuring the dimensions of the doorway to install a screen door • Buying pizzas for a class party • Buying carpeting for a living room floor
  • 83. Estimation Skills • Will the sum of 8 + 7 be greater than or less than 20? Why? • Is the difference of 81 and 29 closer to 40, 50, or 60? Why? • Is the sum of 1/3 + 4/8 greater than or less than 2? Why? • How would you estimate the product of 2.4 and 63? Will the product be between 2 x 60 and 3 x 60? Why or why not?
  • 84. Word Webs Select a word or phrase.
  • 85. Sort and Label
  • 86. Sort and Label • sum, minus, join, compare, subtract, add, take apart, plus • pint, foot, measuring cup, ounce, inch, scale, yard, pound, ruler • square, trapezoid, hexagon, rectangle, rhombus, triangle, pentagon • expression, equation, addition, operation, inequality, comparison, variable, division
  • 87. Mystery Words area perimeter volume length
  • 88. Translate the Symbol • 4 dollars and 10 cents is greater than 4 dollars and 5 cents • One-fourth of 16 is 4 • Doubling a number then adding six more
  • 89. Translate the Symbol • 12 = 7 + 5 • 4 + x = 6 • 3 x 4 > 2 x 5
  • 90. Which Is More Challenging? • 4 dollars and 10 cents is greater than 4 dollars and 5 cents • One-fourth of 16 is 4 • Doubling a number then adding six more • 12 = 7 + 5 • 4 + x = 6 • 3 x 4 > 2 x 5
  • 91. Word Boxes word Definition Real life example Picture Other words
  • 92. Page 102-103
  • 93. Exploring Standard 7 Look for and make use of structure.
  • 94. Understanding the Standard • How do we help students discover patterns in the number system? • What can we do to help students make sense of mathematics through the use of structure?
  • 95. Can you see the pattern? 1/2 = .50 1/3 = .33 1/5 = .20 1/4 = .25 1/6 = .167 1/10 = .10 1/8 = .125 1/12 = .083 1/20 = .05 1/16 = .0625 1/24 = .0467 1/40 = .025 page 107
  • 96. Properties, Patterns, and Functions There were 10 children at the party. How many were boys and how many were girls? Boys Girls 0 10 1 9 2 8 3 7 4 6 5 5 6 4 7 3 8 2 9 1 10 0
  • 97. Perimeter Patterns If there was a row of 50 connected equilateral triangles, what would the perimeter measure?
  • 98. Number Flexibility There was 1 ½ cupcakes left on the plate and Liam and Molly decided they would eat them. How much might each person have eaten? Be ready to justify your answers.
  • 99. How Do We Get There? • Pattern Cover Up • Pattern in the Hundreds Chart or Multiplication Chart • Ratio Tables • Number Lines
  • 100. Pattern Cover Up 2 5 9 14 20
  • 101. Hundreds Chart On a hundred chart, can students explain the vertical and horizontal pattern? Do they see diagonal patterns?
  • 102. Multiplication Chart Can students explain vertical and horizontal patterns? Do they see diagonal patterns? Can they find patterns that explore equivalent fractions or proportions?
  • 103. Ratio Table Margo was painting flowers on the classroom mural. Every flower had 3 leaves. How many leaves were on the mural after 7 flowers had been painted. Number of flowers 1 2 3 4 5 6 7 Number of leaves 3 6 9 12 15 18 21
  • 104. Ratio Table • Each chicken has two legs. How many legs are on 4 chickens? • 2 jars of peanut butter cost $4.50. How much do 8 jars cost? • To make one dozen ice-cream sandwiches, Katie used ¾ gallon of ice cream. How much ice cream did she need for 60 ice cream sandwiches?
  • 105. Number Line
  • 106. page 117-118
  • 107. Exploring Standard 8 Look for and express regularity in repeated reasoning.
  • 108. Understanding the Standard • Why is it important for students to recognize repetition and reason why it is happening? • When you think about repeated reasoning, what do you think about?
  • 109. How Do We Get There? • Exploring Repetition • Finding Shortcuts • Organizing and Displaying Data to Discover Rules • Classroom Investigations
  • 110. Exploring Repetition 2 + 1 = 3 5 + 1 = 6 3 + 1 = 4 6 + 1 = 7 4 + 1 = 5 What do you notice? What conclusion can you draw?
  • 111. Exploring Repetition
  • 112. Investigations to Find Shortcuts • Read first paragraph under heading Investigations to Find Shortcuts - page 124 • Turn and Talk – about your experience with “Memorization”
  • 113. Investigations to Find Shortcuts • What do you notice? • Do you see any pattern? • Did you notice anything interesting about the solution?
  • 114. Organizing and Displaying Data to Discover Rules Alice jumps rope faster then anyone in her class. She can jump 8 times in 4 seconds. How long will it take her to jump 40 times? Justify your answer. Page 127
  • 115. Orchestrating Classroom Investigations to Discover Shortcuts 1. Opportunities for all students to gather data with partners or teams. 2. Creating compilations of class data. 3. Observing compiled data and discussing insights. Page 128
  • 116. Page 130
  • 117. Our “To Do” List • Introduce all teachers to the practice standards. • Provide examples to illustrate the standards. • Encourage ongoing reflection about the standards. – Professional development – Faculty meetings – Book study or PLCs – Grade level teams – Math coaches
  • 118. Let’s Do Some Math!
  • 119. Practice the Mathematical Practices Three different veterinarians each help a total of 63 dogs and cats in a week, but each veterinarian helps a different number of dogs and cats. How many dogs and cats could each veterinarian have helped?
  • 120. Use tangrams to create the Cat…
  • 121. What is the fractional value of your cat’s tail?
  • 122. How can you find the fractional value of each piece?
  • 123. What questions do you have?
  • 124. Updates • NCCTM Conference: Oct 31-Nov 1 • K-2 Mid-Year Assessments • 3-5 Formative Tasks • Vocabulary Document
  • 125. DPI Mathematics Section Kitty Rutherford Elementary Mathematics Consultant 919-807-3841 kitty.rutherford@dpi.nc.gov Denise Schulz Elementary Mathematics Consultant 919-807-3839 denise.schulz@dpi.nc.gov Johannah Maynor Secondary Mathematics Consultant 919-807-3842 johannah.maynor@dpi.nc.gov Vacant Secondary Mathematics Consultant 919-807-3934 Jennifer Curtis K – 12 Mathematics Section Chief 919-807-3838 jennifer.curtis@dpi.nc.gov Susan Hart Mathematics Program Assistant 919-807-3846 susan.hart@dpi.nc.gov
  • 126. For all you do for our students!