High School Mathematics and PARCC Presented by Don Biery Turn to page 1 of your hand out……… BLAH BLAH BLAH BLAH……. Common Core and The Modeling System Math is not a subject to just fill curriculumn…….it is life, it is all around us, hidden in the fabric of nature. ParCC Modeling Questions The real world is what ParCC is after. Life is biology- Biology is Chemistry – Chemistry is application of Physics – physics is the application of Math Math must be an interdisciplinary item Departments must meet to work together….. Classes must be aligned for content to interact i.e. ALG II needs to teach systems of equations before chemistry get to formulae balancing. Math must have labs………… Math Labs = Math Modeling There are no x’s in the world…. We must teach students to be able to problem solve not solve problems. PARCC High Level Blueprints Mathematics Math item counts per form Assess me nt EOY Ite ms Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 Algebra I Geomet ry Math II Algebra II 3 4 2 8 2 8 2 6 2 4 2 6 2 1 1 9 1 9 1 9 1 9 1 9 5 8 8 7 8 5 1 1 1 2 1 2 1 2 1 2 1 4 - - - 1 1 2 3 3 3 3 3 2 3 9 3 6 3 6 3 4 3 3 3 3 3 5 3 4 3 4 3 4 3 4 3 5 Type I 1Point 8 8 6 8 8 1 0 1 0 1 0 1 0 1 0 1 0 1 0 Type I 2 Point 2 2 3 2 2 1 - - - - - - Type II 3 Point 2 2 2 2 2 2 2 2 2 2 2 2 Type II 4 Point 2 2 2 2 2 2 2 2 2 2 3 3 Type III 3 Point 2 2 2 2 2 2 2 2 2 2 2 2 Type III 6 Point 1 1 1 1 1 1 2 2 2 2 3 3 1 0 1 0 9 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 0 1 0 Type II 4 4 4 4 4 4 4 4 4 4 5 5 Type III 3 3 3 3 3 3 4 4 4 4 5 5 Type I 1 Point Type I 2 Point Type I 4 Point E O Y TOT AL PBA/M YA Type I Type I PBA/M YA TOTA L Math I Math III Overview of Task Types • The PARCC assessments for mathematics will involve three primary types of tasks: Type I, II, and III. • Each task type is described on the basis of several factors, principally the purpose of the task in generating evidence for certain sub claims. Task Type I. Tasks assessing concepts, skills and procedures Description of Task Type • • Balance of conceptual understanding, fluency, and application Can involve any or all mathematical practice standards • Machine scorable including innovative, computer-based formats • Will appear on the End of Year and Performance Based Assessment components II. Tasks assessing expressing reasoning, or precision in mathematical mathematical reasoning III. Tasks assessing modeling / scenario (MP.4) applications • Sub-claims A, B and E • Each task calls for written arguments / justifications, critique of statements (MP.3, 6). • Can involve other mathematical practice standards • May include a mix of machine scored and hand scored responses • Included on the Performance Based Assessment component • Sub-claim C • Each task calls for modeling/application in a real-world context or • Can involve other mathematical practice standards • May include a mix of machine scored and hand scored responses • Included on the Performance Based Assessment component • Sub-claim D Real World Application 1. Financial Literacy…..Leads to Better Understanding and true applicability Real World Application 2. Algebra: Why we learn Real Analysis…..Be true to our Students…… Real World Application 3. Geometry: What can I do with this junk? Real World Application Labs and the real world…. 1. Give them real problems to solve 2. Don’t always tell them what tool to use ---ParCC Create a Lab: 1. Reason for Lab (UBD) 2. Lab reporting out? 3.Lab Procedures 4. Possible issues…… November 2013 HS Type Evidence Statement Most Relevant Standards for Mathematical Content Popcorn Inventory Type III 6 Points HS.D.1-1: Solve multi-step contextual problems with degree of difficulty appropriate to the course, requiring application of knowledge and skills articulated in 7.RP.A, 7.NS.3, 7.EE, and/or 8.EE. 7.RP.2: Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relations hips. c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn . d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. 7.RP.3: Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. These standards are major content in the seventh grade based on the PARCC Model Content Frameworks. Most Relevant Standards for Mathematical Practice Item Description and Assessment Qualities Students creating reasoned estimates must be able to reason abstractly and quantitatively in order to build a model of the situation that is accurate enough for the given situation (MP.2 and MP.4). Working with ambiguity is an important part of the modeling skills expected at high school, and this requires students to productively engage with the quantities given in the context. Modeling is a critical component of the high school standards, and this item requires students to create a model from a complex situation to make a real -world estimate given an unscaffolded situation where a model is a useful tool. To make this model, students will have to reason with the given quantities, their units and the proportional relationships between them. This will require students to understand how to use the numbers mathematically, and then be able to periodically check their own understanding of what those numbers mean. This application task requires students to create a reasoned estimate in response to solve a real-world problem. Students must first wrestle with the data displayed on the Popcorn Inventory page . They should recognize that the amounts in the table are decreasing because each day boxes and popcorn seeds are used. This means that students should recognize that they are using about 15 medium boxes each day and about 25-30 small boxes each day. In order to address the amount of popcorn sold over the weekend, students must first create a viable estimate of the number of cups of popcorn sold, then use the ratio cups of popcorn seed:8 cups of popcorn. Students may choose to use this method to estimate the amount of popcorn seed used Sunday through Thursday; however, other methods could be used to determine the amount of popcorn seed used each day. Students using this method must be sure to account for the amount o f popcorn seed used on Sunday because the original information starts from end of day on Sunday. The final estimate requires students to use the current amount of popcorn seed, the amount used Friday and Saturday, and the amount used Sunday through Thursda y in order to estimate the amount of popcorn seeds she should purchase so there are 100200 pounds left over next Friday morning. Note that ratio and proportional relationships are key skills required for college and career readiness, and this item provides a strong application of that content. Unlike traditional multiple choice, it is difficult to guess the correct answer or use a choice elimination strategy. Scoring Rubric for HS.D.1-1 Task is worth 6 points. Task can be scored as 0, 1, 2, 3, 4, 5, or 6. Scoring consists of 2 points for calculation and 4 points for modeling. Structure (6 points total): - Scoring Information - - 2 points for correctly addressing the cups of popcorn seed needed for SundayThursday o 1 calculation point for adequate estimate o 1 modeling point for adequate estimation strategy 3 points for correctly addressing the cups of popcorn seed needed for Friday and Saturday. o 1 modeling point for adequate estimation strategy for addressing two sizes of boxes for both days. o 1 modeling point for accurate use of the proportion of popcorn seed to popcorn o 1 calculation point for adequate estimate 1 point for correctly estimating the amount of popcorn seed that should be ordered o 1 modeling point for adequate estimation strategy (the calculation is not as important as the strategy) Example student response: On Friday and Saturday, they will sell about 500 large boxes (250 + 250 = 500). I found that they sold about 17 medium boxes ( ( – ) and about 30 small boxes ) each day in the table, so they would sell about 68 (2 x 17 for both – days) medium boxes and 120 (2 x 30 for both days) small boxes on Friday and Saturday combined. That means they need to pop:    Large: Medium (approx.): Small (approx.): I added the three amounts of popcorn to find that they will need about 12,400 cups of popcorn over the weekend. Since -cups of popcorn seed makes 8 cups of popcorn, I know that 1 cup of popcorn seed will make 24 cups of popcorn. That means that they need about cups of popcorn seed for Friday and Saturday. So, they will need about 525 cups of popcorn seed for the weekend. According to the table, they used about 80 cups of popcorn seed each of the remaining days of the week ( – ). They will need 80 x 5 or 400 cups of popcorn seed for Sunday-Thursday. I made this list to make sure she buys enough: ( 69.7 cups currrently) 525 cups of popcorn seed for Friday and Saturday 400 cups of popcorn seed for Sunday-Thursday + 100 extra cups to make sure she is between 100 and 200 cups on Friday morning 1,025 cups of popcorn seed to order in the morning NOTE: There are a wide variety of estimation strategies that can receive full credit. HS Type Evidence Statement Brett’s race Type III 3 Points HS.D.2-5: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-level knowledge and skills articulated in A-CED, N-Q, A-SSE.3, A-REI.6, A-REI.12, A-REI.11-2, limited to linear equations and exponential equations with integer exponents. Clarification: A-CED is the primary content; other listed content elements may be involved in tasks as well. Most Relevant Standards for Mathematical Content Most Relevant Standards for Mathematical Practice Item Description and Assessment Qualities A-CED Creating Equations A-CED.A Create equations that describe numbers or relationships 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. This standard is major content in the course based on the PARCC Model Content Frameworks. This item requires students to model the given situation using equations, then students use that model to determine who will win the race and their margin of victory (MP.4). In order to create and interpret these models, students will have to decontextualize and contextualize the information at various points in the solution process to create a mathematical model and then to interpret the meaning and structure of that model (MP.2). Students that choose to use the graph may create another model of the situation, and look for and use structure within that model (MP.7). This application task requires students to use content from widely applicable algebra standards in order to solve a modeling problem with difficulty expected in high school. Students first create equations that model the situation described in the first paragraph. It is important for students to define their variables when creating equations. Then, students reason with their models, and perhaps the graphing tool, to interpret the model and determine the margin of victory. There are a variety of solution methods that students may use to successfully answer Part B. Scoring Rubric for Sample Item HS.D. 2-5 Scoring Information Task is worth 3 points. Task can be scored as 0, 1, 2, or 3. Task has 2 parts. Scoring for Part A – Formulating the Model – 1 point Student produces two equations to determine the distance in meters from the starting line, of each person as a function of the time x, in seconds since the Olympian starts running. For example, Brett’s distance y, as related to time, x: 3 𝑦 = 8 1 𝑥 + 20 . Or y = 12 100 x + 20 The Olympian’s distance y, as related to time, x: 𝑦 = 10 . NOTE: All variables should be defined. The student may choose to define x as time in seconds since Brett starts running. Scoring for Part B Student earns 1 calculation point for stating the correct winner and the correct margin of victory. Students earn 1 modeling point for providing an accurate justification using the equations in Part A. Sample Student Response 1: • For Brett, 𝑦 = 100 when 100 = 8 1 𝑥 + 20 13 80 = 8 3 𝑥 𝑥 = 9.6 • For the Olympian 𝑦 = 100 when 100 = 10𝑥 . 𝑥 = 10 • So, Brett wins the race by seconds. 10 – 9.6 = 0.4 Sample Student Response 2 : • When Brett finishes the race at 9.6 seconds, the Olympian is only 10(9.6) = 96 meters from the start. Therefore, Brett was 4 meters ahead of the Olympian when he finished the race. Note: • If Part A contains incorrect equations, but Part B is correct based on one or two incorrect equations in Part A, the student is still awarded 1 or 2 points of the 3 possible points. Task score: The task score is the sum of the points awarded in each component. Lessons Learned from OOOPPPSSS  1. Always do Lab Your Self…..  2. Don’t always give them the tools, allow them to ask for them  3. Evaluate effectiveness of lab  4. Evaluate effectivness of reporting (group participation) ParCC and CCSS Curriculumn  Discussion: Groups 5 ways to aid in implementation  5 hurdles you see  5 ways to overcome those obstacles. Questions ???????  Contact Information  Don Biery  [email protected]