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“Teachers are thus free to provide students with whatever tools and knowledge their professional judgment and experience identify as most helpful for meeting the goals set out in the Standards.” ~ Introduction to the CCSS HS Algebra February 2013 Outcomes Align the regional/district Algebra course to the PARCC framework Create tape diagrams and double number lines to solve application problems Explain the information we have, need and will make do with PARCC Resources Progressions Illustrative Mathematics http://commoncoretools.wordpress.com/ Quality Review Rubrics http://illustrativemathematics.org/ Common Core Tools http://ime.math.arizona.edu/progressions/ http://www.achieve.org/files/TriState-MathematicsQuality-RubricFINAL-May2012.pdf Achieve the Core http://www.achievethecore.org/ PARCC Components Key Advances from the Previous Grade Discussion of Mathematical Practices in Relation to Course Content Fluency Recommendations Pathway Summary Tables Assessment Limits Tables Math Practice Meditation Imagine your best students… consider how they showed each of these qualities… 1. Perseverance 2. Reason abstractly and quant. 3. Construct and critique 4. Model 5. Use tools strategically 6. Precision 7. Use structure 8. Find and express repeated reasoning Look For’s in a CCLS Lesson Fluency Task (~10 mins) Modeling Concept Building Application Debrief Pair Sharing Exit Ticket (Daily formative assessment) Module Sources PK-5 Common Core Inc 6-12 CCI EduTron Construct Scope and Sequence PK-8 Grade Level Map Modules Lessons Assessment Mid Module Assessment New Module End of Module Assessment Content Gap Instruction Multiplication Division Strategies Algebraic Understanding Fractions Number Sense & Place Value Quiz 1 What teaching materials will likely be available? Is State Ed is making and providing all of the math materials teachers need? What are important documents to help build our HS curriculum? Math Modules PK-5 6-12 Common Core Inc CCI with Support from EdutTron Fluency Application Conceptual Fluency Application Conceptual Fluency Required Fluencies Grade Required Fluency K Add/Subtract within 5 1 +/- within 10 2 Add/Subtract within 20 Add/Subtract within 100 (paper and pencil) 3 Multiply/divide within 100 Add/Subtract within 1000 4 Add/Subtract within 1,000,000 5 Multi-digit multiplication 6 Multi-digit division Multi-digit decimal operations 7 Solve px+q=r, p(x+q)=r 8 Solve simple 2x2 systems by inspection Application Multiplication Facts Conceptual Fluency X 1 2 3 4 5 6 7 8 9 1 1x1 1x2 1x3 1x4 1x5 1x6 1x7 1x8 1x9 2 2x1 2x2 2x3 2x4 2x5 2x6 2x7 2x8 2x9 3 3x1 3x2 3x3 3x4 3x5 3x6 3x7 3x8 3x9 4 4x1 4x2 4x3 4 x 4 4x5 4x6 4x7 4x8 4x9 5 5x1 5x2 5x3 5x4 5x5 5x6 5x7 5x8 5x9 6 6x1 6x2 6x3 6x4 6x5 6x6 6x7 6x8 6x9 7 7x1 7x2 7x3 7x4 7x5 7x6 7x7 7x8 7x9 8 8x1 8x2 8x3 8x4 8x5 8x6 8x7 8x8 8x9 9 9x1 9x2 9x3 9x4 9x5 9x6 9x7 9x8 9x9 Commutative Property Identity Property Doubles Squares Fives Challenge (benchmark) (benchmark) Gene Jordan’s work but I got the Idea from Gina King’s article:www.nctm.org teaching children mathematics • King, Fluency with Basic Addition, September 2011 p. 83 Application Addition Facts Conceptual Fluency Gina King’s article:www.nctm.org teaching children mathematics • King, Fluency with Basic Addition, September 2011 p. 83 Application Conceptual Fluency Example Finger Counting 1,2,3, sit on 10 High 5 Fluency Application Conceptual Fluency Fluency Fast Frequent Fun Application Conceptual Fluency Math Sprints Ready, Set, Go! Sprint A Review Sprint A Math Moves Sprint B Review Sprint B Cool Down Conceptual Modeling Application Conceptual Fluency Conceptual Concrete Pictorial Abstract Moving both ways Draw a picture of 4+4+4 Show your thinking Explain, defend and critique the reasoning of others Application Conceptual Fluency Concrete Model Equation X+3=5 Application Conceptual Fluency Tape Diagram Problems Tape diagrams are best used to model ratios when the two quantities have the same units. Tape Diagrams: Q1 1. David and Jason have marbles in a ratio of 2:3. Together, they have a total of 35 marbles. How many marbles does each boy have? Tape Diagrams : Q2 2. The ratio of boys to girls in the class is 5:7. There are 36 children in the class. How many more girls than boys are there in the class? Tape Diagrams Q3: Comparing 3 items Lisa, Megan and Mary were paid $120 for babysitting in a ratio of 2: 3: 5. How much less did Lisa make than Mary? Tape Diagrams Q4: Different Ratios The ratio of Patrick’s M & M’s to Evan’s is 2: 1 and the ratio of Evan’s M & M’s to Michael’s is 4: 5. Find the ratio of Patrick’s M & M’s to Michael’s. Tape Diagrams Q5: Changing Ratios The ratio of Abby’s money to Daniel’s is 2: 9. Daniel has $45. If Daniel gives Abby $15, what will be the new ratio of Abby’s money to Daniel’s? Double Number Line Double number line diagrams are best used when the quantities have different units. Double number line diagrams can help make visible that there are many, even infinitely many, pairs of numbers in the same ratio—including those with rational number entries. As in tables, unit rates (R) appear in the pair (R, 1). Double Number Line: Finding average rate It took Megan 2 hours to complete 3 pages of math homework. Assuming she works at a constant rate, if she works for 8 hours, how many pages of math homework will she complete? What is the average rate at which she works? Identify properties of the RDW modeling technique for application problems Read (2x) Draw a model Write an equation or number sentence Write and answer statement Unit Object Context Use RDW to solve Problem Modeling Challenge 2 boxes of salt and a box of sugar cost $6.60. A box of salt is $1.20 less than a box of sugar. What is the cost of a box of sugar? Salt Salt $6.60 3 parts = $6.60- $1.20 Sugar $1.20+$1.80= $3.00 $1.20 3 parts = $5.40 1 part = $5.40 ÷ 3 = $1.80 Challenging Problems The students in Mr. Hill’s class played games at recess. Mika Said: “Four 6 boys played soccer 4 girls played soccer 2 boys jumped rope 8 girls jumped rope Chaska Said: “For every girl that played soccer, two girls jumped rope.” more girls jumped rope than played soccer.” Mr Hill Said: “Mika compared girls by looking at the difference and Chaska compared the girls using a ratio” 1) Compare the number of boys who played soccer and jumped rope using the difference. Write your answer as a sentence as Mika did. 2) Compare the number of boys who played soccer and jumped rope using a ratio. Write your answer as a sentence as Chaska did. 3) Compare the number of girls who played soccer to the number of boys who played soccer using a ratio. Write your answer as a sentence as Chaska did. Challenging Problems Compare 3 4 these fractions: and 3+1 4+1 Which one is bigger than the other? Why? Application Application Conceptual Fluency Application The beginning of the year is characterized by establishing routines that encourage hard, intelligent work through guided practice rather than exploration. Slower and deeper Use the Read-Draw-Write (RDW) steps Application Conceptual Fluency Application Wrap up Thanks for coming! Links www.btboces2.org/mathpd http://www.parcconline.org/samples/mat hematics/grade-6-slider-ruler http://www.parcconline.org/samples/mat hematics/grade-7-mathematics www.Engageny.org High School Functions A--‐REI.4. Solve quadratic equations in one variable. Seeing Structure in a Quadratic Equation High School Illustrative Sample Item 43 A--‐REI.4. Solve quadratic equations in one variable. Seeing Structure in a Quadratic Equation High School Illustrative Sample Item 44 A--‐SSE, Seeing Structure in Expressions Aligns to the Standards and Reflects Good Practice High School Sample Illustrative Item: Seeing Structure in a Quadratic Equation Task Type I: Tasks assessing concepts, skills and procedures Alignment: Most Relevant Content Standard(s) A-REI.4. Solve quadratic equations in one variable. a) Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b) Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a bi for real numbers a and b. Alignment: Most Relevant Mathematical Practice(s) Students taking a brute-force approach to this task will need considerable symbolic fluency to obtain the solutions. In this sense, the task rewards looking for and making use of structure (MP.7). 45 Aligns to the Standards and Reflects Good Practice High School Illustrative Item Key Features and Assessment Advances The given equation is quadratic equation with two solutions. The task does not clue the student that the equation is quadratic or that it has two solutions; students must recognize the nature of the equation from its structure. Notice that the terms 6x – 4 and 3x – 2 differ only by an overall factor of two. So the given equation has the structure 2 = 2 where Q is 3x – 2. The equation Q2 - 2Q is easily solved by factoring as Q(Q-2) = 0, hence Q = 0 or Q = 2. Remembering that Q is 3x – 2, we have 3 − 2 = 0 or 3 − 2 = 2. These two equations yield the solutions 23 and 43. Unlike traditional multiple-choice tests, the technology in this task prevents guessing and working backwards. The format somewhat resembles the Japanese University Entrance Examinations format (see innovations in ITN Appendix F). A further enhancement is that the item format does not immediately indicate the number of solutions. 46 Gr. 7 Ratios and Proportions: Equivalent Ratios and Fractions Gr. 7 Ratios and Proportions: Equivalent Ratios and Fractions Gr. 7 Ratios and Proportions: Equivalent Ratios and Fractions Gr. 7 Ratios and Proportions: Equivalent Ratios and Fractions “Teachers are thus free to provide students with whatever tools and knowledge their professional judgment and experience identify as most helpful for meeting the goals set out in the Standards.” ~ Introduction to the CCSS Math Sprints Fluency in a minute