“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 • http://ime.math.arizona.edu/progressions/ • Illustrative Mathematics • http://illustrativemathematics.org/ • Common Core Tools • http://commoncoretools.wordpress.com/ • Quality Review Rubrics • http://www.achieve.org/files/TriState-Mathematics-Quality-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… Perseverance Reason abstractly and quant. Construct and critique Model Use tools strategically Precision Use structure 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)
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?
Multiplication Facts 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
Addition Facts Gina King’s article:www.nctm.org teaching children mathematics • King, Fluency with Basic Addition, September 2011 p. 83
Fluency Example • Finger Counting • 1,2,3, sit on 10 • High 5
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
Concrete Model Equation X + 3 = 5
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
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 3 parts = $5.40 1 part = $5.40 ÷ 3 = $1.80 $1.20+$1.80= $3.00 $1.20
Challenging Problems Mika Said: “Four more girls jumped rope than played soccer.” Chaska Said: “For every girl that played soccer, two girls jumped rope.” • The students in Mr. Hill’s class played games at recess. • 6 boys played soccer • 4 girls played soccer • 2 boys jumped rope • 8 girls jumped rope • 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. Mr Hill Said: “Mika compared girls by looking at the difference and Chaska compared the girls using a ratio”
Challenging Problems • Compare these fractions: Which one is bigger than the other? Why? and
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
Wrap up Thanks for coming! Links • www.btboces2.org/mathpd • http://www.parcconline.org/samples/mathematics/grade-6-slider-ruler • http://www.parcconline.org/samples/mathematics/grade-7-mathematics • www.Engageny.org
High School Functions A--‐REI.4. Solvequadraticequationsinonevariable.
High School Illustrative Sample Item Seeing Structure in a Quadratic Equation A--‐REI.4. Solvequadraticequationsinonevariable.
High School Illustrative Sample Item Seeing Structure in a Quadratic Equation A--‐SSE, SeeingStructureinExpressions
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. 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. 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 abi 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).
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 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 . These two equations yield the solutions and . 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.