Table of Contents : State of Oregon



Oregon Adult College and Career Standards MathematicsModule 1Workbook1376060206042Table of ContentsActivating Prior KnowledgePage 3Reflection 1 ................................................................................................Page 4An Application Problem Using Standard 4.NF.4cPage 5Reflection 2: The Use of Area ModelsPage 6Directions: Engaging the Three Components of RigorPage 7Worksheet: Engaging the Three Components of RigorPage 8Answer Key: Engaging the Three Components of RigorPage 11 HYPERLINK \l "_bookmark8" Wrapping it Up: Thinking About All Three Shifts.Page 14References Used in Mathematics Module 1 PresentationPage 15Lesson Plans and other Valuable Links.Page 17The following workbook is the companion to the OACCRS Mathematics Module 1 presentation. You may print out the workbook or complete it digitally.Because this training module may be used in several different settings with different requirements, please check with your director for specific instructions about completing and turning in responses for the activities in this workbook.Activating Prior KnowledgeYou may download and print this worksheet or input your answers on a copy saved locally under your name.How does your instruction differ from the way you were taught?What does a shift in instruction mean in your teaching?What is meant by the term “shifts in content” in your context?Reflection 1Consider your thoughts and briefly respond. Feel free to use additional paper as necessary.How would the instruction for the OACCRS example below vary from the standard “Multiply a fraction by a whole number” ?Solve word problems involving multiplication of a fraction by a whole number by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? (4.NF.4c)An Application Problem Using Standard 4.NF.4cThe highway department needs to replace the guard rails along the eight mile stretch of Redwood Highway between Cave Junction and Kerby. They know they can install 9/10 of a mile of rails every day. Can the highway department get the job done at that rate in 9 work days?What different diagrams could be used to model this situation?What other “hands-on” models (besides diagrams) could be used to model this problem? What would be a logical equation or inequality to model this situation?831956833609In each row, the highlighted cells represent the amount of railing installed in 1 day.ONE WAY OF THINKING (not necessarily the BEST or ONLY way of thinking):Each of the nine days, the railing installed is one tenth short of a mile. After nine days they would have installed nine tenths of a mile short of nine miles. Since nine tenths of a mile is less than 1 mile, 9 miles minus nine tenths would be more than 9-1, or more than 8, so they could get all the railing installed in 9 days.Other ways to model the problem might include using a centimeter ruler or measuring tape where each centimeter corresponds to one tenth of a mile or using Cheezits or other like food items to represent each tenth of a mile. Can you think of a better way to model the problem so students can easily justify their solution?One logical inequality modeling this problem would be 9 (1) -9(1/10) > 8, but there are many other possible (and perhaps more obvious) inequalities and equations.5Reflection 2: The Use of Area ModelsPlease evaluate the merit of using an area model for multiplication of common fractions and/or decimal fractions.Please include in your brief answer both the benefits and the drawbacks of using the area model for multiplication of common fractions and/or decimal fractions.Workshop Materials Mathematics Foundational Unit 3 Engaging the Three Components of RigorDirections for Engaging the Three Components of RigorCheck the component(s) of rigor that are likely to be required in a lesson, activity, or task that targets each CCR Standard on the Engaging the Three Components of Rigor worksheet.Make notes about your rationales.Talk about our reasoning with a colleague or two, using these questions to guide your discussion:What makes you think a particular component of rigor applies?Are there certain words or phrases in the standard that provide clues?Which components of rigor might appear together in a single standard? Explain.Which components of rigor are not likely to appear together in a single standard? Explain.Here is an example:Given: Understand that a set of data collected to answer a statistical question has a distribution, which can be described by its center, spread, and overall shape.(6.SP.2; Level C)A Sample Response: The focus of this standard is on the conceptual understanding of the variability of quantitative data. (Key word: understand)Note: Depending on the particular lesson, this standard could be used to develop a different component of rigor. For example, if the lesson deals with the COVID-19 pandemic and the task is explain why “flattening the curve” for spread of the virus is important, it would fall under the category of application.Implementing College and Career Readiness Standards in Adult Education - page 1Workshop Materials Mathematics Foundational Unit 3 Engaging the Three Components of RigorCoding Guide:Worksheet: Engaging the Three Components of RigorCU = Conceptual Understanding PSF = Procedural Skill and Fluency A = ApplicationNote: More than one component of rigor may apply for a R StandardComponent ofRigorRationale1Understand that a set of data collected to answer a statistical question has a distribution, which can be described by its center, spread, and overall shape. (6.SP.2; Level C)CUPSFA2Fluently multiply multi-digit whole numbers using the standard algorithm. (5.NBT.5; Level C)CUPSFA3Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement. (4.MD.5; Level C)CUPSFAWorkshop Materials Mathematics Foundational Unit 3 Engaging the Three Components of Rigor4Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. (6.SP.2; Level C)CUPSFA5Solve multi-step word problems posed with whole numbers and having whole number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (4.OA.3; Level C)CUPSFA6Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. (7.NS.2d; Level D)CUPSFA7Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. (6.NS.1; Level C)CUPSFAWorkshop Materials Mathematics Foundational Unit 3 Engaging the Three Components of Rigor8Understand solving an equation or inequality as a process of answering a question: Which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. (6.EE.5; Level C)CUPSFA9Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real-world problems. (5.MD.5c; Level C)CUPSFA10Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. (5.NF.3; Level C)CUPSFAWorkshop Materials Mathematics Foundational Unit 3 Engaging the Three Components of RigorCoding Guide:Answer Key: Engaging the Three Components of RigorCU = Conceptual Understanding PSF = Procedural Skill and Fluency A = ApplicationNote: Some of the standards used in this activity also were used in the coherence unit, and so they should look familiar. Some standards may address more than one component of R StandardComponent ofRigorRationale1Understand that a set of data collected to answer a statisticalCUPSFAThe focus of this standard is on thequestion has a distribution, which can be described by its center,spread, and overall shape. (6.SP.2; Level C)√----conceptual understanding of the variabilityof quantitative data.2Fluently multiply multi-digit whole numbers using the standard algorithm. (5.NBT.5; Level C)CUPSFAThis standard focuses on fluency with multiplication of whole numbers.√--3Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts ofangle measurement. (4.MD.5; Level C)CUPSFAThis standard focuses on developing an understanding of angle measurement.√----4Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. (6.SP.2; Level C)CUPSFAThis standard focuses on developing an understanding of ratios.√----Workshop Materials Mathematics Foundational Unit 3 Engaging the Three Components of RigorCCR StandardComponent ofRigorRationale5Solve multi-step word problems posed with whole numbers andCUPSFAThis standard asks for understanding thehaving whole number answers using the four operations, includingproblems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.(4.OA.3; Level C)√--√concept of remainders and expressions, butalso for applications that use them.6Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. (7.NS.2d; Level D)CUPSFAThis standard focuses on the procedural skill of converting fractions to decimals. (Note: This example does not use the word“fluent.”)--√--7Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent theproblem. (6.NS.1; Level C)CUPSFAThis standard asks for understanding (to interpret), and also applications that require division of fractions.√--√8Understand solving an equation or inequality as a process of answering a question: Which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes anequation or inequality true. (6.EE.5; Level C)CUPSFAThis standard focuses on developing an understanding of solving an equation or inequality.√----9Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real-world problems. (5.MD.5c; Level C)CUPSFAThis standard asks for understanding of the additive quality of volume and also solving problems involving volume.√--√Workshop Materials Mathematics Foundational Unit 3 Engaging the Three Components of RigorCCR StandardComponent ofRigorRationale10Interpret a fraction as division of the numerator by theCUPSFAThis standard asks for understanding of adenominator (a/b = a ÷ b). Solve word problems involvingdivision of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. (5.NF.3; Level C)√--√fraction as a division problem (and theconnection between remainders and solutions as mixed numbers), and also solving problems involving division withmixed number solutions.The compound standards in 5, 7, 9, and 10 are likely to have two components of rigor present in a lesson.Wrapping It Up: Thinking About All Three Shifts(Use additional paper, if necessary.)Look again at the screen shot of part of a typical comparing fractions worksheet found on slide 70 and at the non-routine problem to its right.Try the problem making a fraction as close to one as you can using the given parameters, then respond (briefly) to the points below.What skills you are using (as well as those your students might use) in working on the problemHow the problem could be modified for students at different mathematics levelsWhat tools would help “open up” the problemHow the key shifts of focus, cohesion, and rigor can be addressed with this taskReferencesACT National Curriculum Survey (2016). Education and work in a time of change. Retrieved from (2011, October 11). Seven myths about rigor. Retrieved from , A. (2015). Mathematics for what? High school students reflect on mathematics as a tool for social inquiry. Democracy and Education, 23 (1). Retrievedfrom 1–11Curry, D. (2017). Where to focus so students become college and career ready? Journal of Research and Practice for Adult Literacy, Secondary and Basic Education, 6(1), 62-66.Retrieved from, K., Stigler, J., & Thompson, B. (2011). What community college developmental mathematics students understand about mathematics, Part II: The interviews. MathAMATYC educator, 2(3), 4-18. Retrieved from , B. (2017, July 23). The structure is the standards. Retrieved from Center for Educational Statistics (2015). Trends in international mathematics and science study. Retrieved from Higher Education Coordinating Committee (2019). Oregon adult college and career readiness: Mathematics handbook. Retrieved from programs/ccwd/Documents/Mathematics%20Standards%20Handbook.pdf.pdfRivera, Connie, (2015). College and career readiness standards progressions of mathematical domains. Retrieved from andards_-_progression_of_mathematical_domains.pdfSchmidt, W.H., H.A. Wang, and C.C. McKnight, (2005) Curriculum coherence: An examination of US mathematics and science content standards from an international perspective.Journal of Curriculum Studies, 37(5). doi: 10.1080/0022027042000294682Steen, L.A. (2007). The future of mathematics education. ASCD Curriculum Handbook Mathematics, 65(3). Retrieved from curriculum- handbook/409/chapters/The-Future-of Mathematics-Education.aspxTargeting aspects of rigor in math instruction. (n.d.). Retrieved May 15, 2020, from U.S. Department of Education (2016). College and career readiness standards in-action: Workshop materials mathematics. Retrieved from 1Materials/Math1 part_mat.pdfVickers, A. & Rivera, C (2016). Uncovering coherence using area models [Video file]. Retrieved from , H. (2011). The mis-education of mathematics teachers. Notices of the AMS, 58(3).Retrieved from , J. (2011). Examples of structure in the common core state standards: standards for mathematical content. Retrieved from content/uploads/2011/07/ccssatlas_2011_07_06_0956_p1p2.pdfLesson Plans and Other Valuable LinksAs stated on slide 21 in the presentation, resources for the Oregon Adult College and Career Standards (OACCRS) are relatively easy to locate online. It is important that you realize the link between OACCRS and the Common Core State Standards (CCSS). The Oregon Adult College and Career Standards are derived from the Common Core State Standards. The coding was purposely retained.One standard illustrated in the PowerPoint was 4.NF.4c. The most important thing to understand about the code is that the leading 4 means that the objective can be found in the Grade 4 materials. (The NF means that it came from the number and operations – fractions domain, and 4c means that it is the fourth standard in that domain, but instructors do not need to know that to use the coding.) Probably the easiest way to use the coding is to simply copy and paste it into a browser. Below are screen shots of the first of more than 22,000,000 hits for that code. (The screen shots are too small to read, but added here for illustration.)126322311538412442282537153As you might be able to see there are many resources available, but of course, some of them are better than others. Sometimes, especially if the standard seems abstruse, simply viewing how others have handled the content is helpful.A word of caution: Teaching in Adult Basic Skills is quite different from teaching in a K-12 setting, and instructors need to be very sensitive about that. It will probably be insulting to adults to show a video titled “Grade 4: Number and Operations: Fractions.” Sometimes worksheets have grade levels embedded in a header or footer – hurray for Adobe editing, white-out, or scissors!Following are links to general mathematics resources that can help you find on-target lessons, many of which have open licenses so they can be used as is or modified to suit the needs. Other links have great general mathematics instruction ideas.EngageNY This site is very comprehensive and includes detailed lesson plans with interactive learning experiences, guided practice, independent practice, and assessments. The link here is for Grade 5 Mathematics, Module 1 (Place Value and Decimal Fractions). There are 5 other modules in the Grade 5 mon Core Resources – Lesson plans and instructional ideas presented by grade level and domain. Use the leading digit in the OACCRS code to identify a grade level, then a domain (the letters that follow the leading digit) menu pops upKhan Academy Common Core Map: It’s a challenge to find specific lessons that align with Common Core proficiencies. Khan Academy’s Common Core Map takes the guess work out that, providing an overview of processes and proficiencies with links to their aligned, web-based lessons. Khan’s Common Core resource page is another great source.Free Lessons from K-5 Math Teaching Resources: This is another site that has taken the guess work out of finding Common Core process-aligned resources. Here, you’ll find a comprehensive pool of K-5 math resources organized by grade level, and all are available as PDF download.Teaching Channel Common Core Video Series: The Teaching Channel’s videos provide an amazing in-classroom look at the Common Core in practice. For grades K-5, Teaching Channel has produced more than 70 videos, covering both math and English lessons. You’ll also find some great overview videos that will help answer general questions about implementing the new standards.Illustrative Mathematics: This link is for Grade 4, Number and Operations Base 10. To change the grade level, use the arrows on the green title bar. The page will show the specific standard and has clickable links to related activities including commentary for instructors.Learn Zillion – The link here is for lesson on area and surface area. This site is connected to Illustrative Mathematics (above) and it has short instructional videos and lesson plans. Many, but not all of the resources on this site are free, but instructors must register to access them.Mathispower4u has over 2,600 video tutorials on topics from arithmetic to calculus. His website and videos feature both math lessons and examples, and many of the videos have been incorporated into online homework questions available at . The CC BY SA license lets others remix, adapt, and build upon your work as long as credit is cited and the user licenses new creations under the identical terms.Growing Math Roots is an open resource compiled by two of Oregon’s Adult Basic Skills Mathematics Trainers. It is closely aligned with the OACCRS and suggests a variety of types of lessons aimed at middle level adult learners.Virtual Nerd is an expansive group of instructional videos that clearly explain mathematics concepts. This site, except for the initial title page, does not indicate a grade level. The videos are very process oriented, so instructors will have to incorporate application, modeling, group interaction, and so on. ................
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