Planning Advice Focus on the Clusters, The mathematics standards call for a greater focus Rather than racing to cover topics in today s mile wide inch deep. curriculum we need to use the power of the eraser and significantly narrow and deepen how time and energy is spent in. the mathematics classroom There is a necessity to focus deeply on the major work of each grade to enable students to. gain strong foundations solid conceptually understanding a high degree of procedural skill and fluency and the ability. to apply the mathematics they know to solve problems both in and out of the mathematics classroom. www achievethecore org, As the Kansas College and Career Ready Standards KCCRS are carefully examined there is a realization that with time. constraints of the classroom not all of the standards can be done equally well and at the level to adequately address the. standards As a result priorities need to be set for planning instruction and assessment Not everything in the. Standards should have equal priority Zimba 2011 Therefore there is a need to elevate the content of some. standards over that of others throughout the K 12 curriculum. When the Standards were developed the following were considerations in the identification of priorities 1 the need to. be qualitative and well articulated 2 the understanding that some content will become more important than other 3. the creation of a focus means that some essential content will get a greater share of the time and resources While the. remaining content is limited in scope 4 a lower priority does not imply exclusion of content but is usually intended. to be taught in conjunction with or in support of one of the major clusters. The Standards are built on the progressions so priorities have to be chosen with an eye to the. arc of big ideas in the Standards A prioritization scheme that respects progressions in the. Standards will strike a balance between the journey and the endpoint If the endpoint is. everything few will have enough wisdom to walk the path if the endpoint is nothing few will. understand where the journey is headed Beginnings and the endings both need particular care. It would also be a mistake to identify such standard as a locus of emphasis Zimba 2011. The important question in planning instruction is What is the mathematics you want the student to walk away with. In planning for instruction grain size is important Grain size corresponds to the knowledge you want the student to. know Mathematics is simplest at the right grain size According to Daro Teaching Chapters Not Lessons Grain Size of. Mathematics strands are too vague and too large a grain size while lessons are too small a grain size About 8 to 12. units or chapters produce about the right grain size In the planning process staff should attend to the clusters and. think of the standards as the ingredients of cluster while understanding that coherence exists at the cluster level across. A caution Grain size is important but can result in conversations that do not advance the intent of this structure. Extended discussions that argue 2 days instead of 3 days on a topic because it is a lower priority detract from the overall. intent of suggested priorities The reverse is also true As Daro indicates lenses focused on. lessons can also provide too narrow a view which compromises the coherence value of. closely related standards, The video clip Teaching Chapters Not Lessons Grain Size of Mathematics that follows presents Phil Daro further. explaining grain size and the importance of it in the planning process Click on photo to view video. Along with grain size clusters have been given priorities which have important implications for instruction These. priorities should help guide the focus for teachers as they determine allocation of time for both planning and instruction. The priorities provided help guide the focus for teachers as they demine distribution of time for both planning and. instruction helping to assure that students really understand before moving on Each cluster has been given a priority. level As professional staffs begin planning developing and writing units as Daro suggests these priorities provide. guidance in assigning time for instruction and formative assessment within the classroom. Each cluster within the standards has been given a priority level by Zimba The three levels are referred to as Focus. Additional and Sample Furthermore Zimba suggests that about 70 of instruction should relate to the Focus clusters. In planning the lower two priorities Additional and Sample can work together by supporting the Focus priorities The. advanced work in the high school standards is often found in Additional and Sample clusters Students who intend to. pursue STEM careers or Advance Placement courses should master the material marked with within the standards. These standards fall outside of priority recommendations. Appropriate Use, Recommendations for using cluster level priorities. Use the priorities as guidance to inform instructional decisions regarding time and resources spent on clusters. by varying the degrees of emphasis, Focus should be on the major work of the grade in order to open up the time and space to bring the Standards. for Mathematical Practice to life in mathematics instruction through sense making reasoning arguing and. critiquing modeling etc, Evaluate instructional materials by taking the cluster level priorities into account The major work of the grade. must be presented with the highest possibility quality the additional work of the grade should indeed support. the Focus priorities and not detract from it, Set priorities for other implementation efforts taking the emphasis into account such as staff development new. curriculum development revision of existing formative or summative testing at the state district or school level. Things to Avoid, Neglecting any of the material in the standards rather than connecting the Additional and Sample clusters to the. other work of the grade, Sorting clusters from Focus to Additional to Sample and then teaching the clusters in order To do so would. remove the coherence of mathematical ideas and miss opportunities to enhance the focus work of the grade. with additional clusters, Using the clusters headings as a replacement for the actual standards All features of the standards matter. from the practices to surrounding text including the particular wording of the individual content standards. Guidance for priorities is given at the cluster level as a way of thinking about the content with the necessary. specificity yet without going so far into detail as to comprise and coherence of the standards grain size. Depth Opportunities, Each cluster at a grade level and each domain at the high school identifies five or fewer standards for in depth. instruction called Depth Opportunities Zimba 2011 Depth Opportunities DO is a qualitative recommendation about. allocating time and effort within the highest priority clusters the Focus level Examining the Depth Opportunities by. standard reflects that some are beginnings some are critical moments or some are endings in the progressions The. DO s provide a prioritization for handling the uneven grain size of the content standards Most of the DO s are not small. content elements but rather focus on a big important idea that students need to develop. DO s can be likened to the Priorities in that they are meant to have relevance for instruction assessment and. professional development In planning instruction related to DO s teachers need to intensify the mode of engagement. by emphasizing tight focus rigorous reasoning and discussion and extended class time devoted to practice and. reflection and have high expectation for mastery See Table 7 Appendix Depth of Knowledge DOK. In this document Depth Opportunities are highlighted pink in the Standards section For example. 5 NBT 6 Find whole number quotients of whole numbers with up to four digit dividends and two digit divisors using. strategies based on place value the properties of operations and or the relationship between multiplication and division. Illustrate and explain the calculation by using equations rectangular arrays and or area models. Depth Opportunities can provide guidance for examining materials for purchase assist in professional dialogue of how. best to develop the DO s in instruction and create opportunities for teachers to develop high quality methods of. formative assessment,Standards for Mathematical Practice in Grade 3. The Common Core State Standards for Mathematical Practice are practices expected to be integrated into every mathematics lesson for all students. Grades K 12 Below are a few examples of how these Practices may be integrated into tasks that Grade 2 students complete. Practice Explanation and Example, 1 Make Sense Mathematically proficient students in Grade 3 examine problems can make sense of the meaning of the task and find an entry point or a way to. and Persevere start the task Grade 3 students also develop a foundation for problem solving strategies and become independently proficient on using those. in Solving strategies to solve new tasks They might use concrete objects or pictures to show the actions of a problem If students are not at first making. Problems sense of a problem or seeing a way to begin they ask questions that will help them get started They are expected to persevere while solving. tasks that is if students reach a point in which they are stuck they can reexamine the task in a different way and continue to solve the task. Students in Grade 3 complete a task by asking themselves the question Does my answer make sense Example to solve a problem involving. multi digit numbers they might first consider similar problems that involve multiples of ten or one hundred Once they have a solution they look. back at the problem to determine if the solution is reasonable and accurate They often check their answers to problems using a different. method or approach, 2 Reason Mathematically proficient students in Grade 3 recognize that a number represents a specific quantity They connect the quantity to written. abstractly and symbols and create a logical representation of the problem at hand considering both the appropriate units involved and the meaning of the. quantitatively quantities This involved two processes decontextualizing and contextualizing In Grade 3 students represent situations by decontextualizing. tasks into numbers and symbols For example to find the area of the floor of a rectangular room that measures 10 ft by 12 ft a student might. represent the problem as an equation solve it mentally and record the problem and solution as 10 x 12 120 She has decontextualized the. problem When she states at the end that the area of the room is 120 square feet she has contextualized the answer i order to solve the original. problem Problems like this that begin with a context and are then represented with mathematical objects or symbols are also examples of. modeling with mathematics SMP 4, 3 Construct Mathematically proficient students in Grade 3 accurately use definitions and previously established solutions to construct viable arguments about. viable mathematics Grade 3 students might construct arguments using concrete referents such as objects pictures and drawings They refine their. arguments and mathematical communication skills as they participate in mathematical discussions involving questions like How did you get that and Why is. critique the that true They explain their thinking to others and respond to others thinking For example when comparing the unit fractions 1 3 and 1 5. reasoning of students may generate their own representation of both fractions and then critique each other s reasoning in class as they connect their. others arguments to the representations that they created Students in Grade 3 present their arguments in the form of representations actions on. those representations and explanations in words oral and written. 4 Model with Mathematically proficient students in Grade 3 experiment with representing problem situations in multiple ways including numbers words. mathematics mathematical language drawing pictures using objects acting out making a chart list or graph creating equations etc They model real life. mathematical situations with a number sentence or an equation and check to make sure that their equation accurately matches the problem. context Students should have ample opportunities to connect the different representations and explain the connections Grade 3 students. should evaluate their results in the context of the situation and reflect on whether the results make sense. 5 Use Mathematically proficient students in Grade 3 consider the available tools including estimation when solving a mathematical problem and. appropriate decide when certain tools might be helpful The tools that students in Grade 3 might use physical objects place value base ten blocks. tools hundreds charts number lines tape diagrams fraction bars arrays tables graphs and concrete geometric shapes e g pattern blocks 3 d. strategically solids paper and pencil rulers and other measuring tools grid paper virtual manipulatives and concrete geometric shapes e g pattern blocks. 3 d solids etc Students should also have experiences with educational technologies such as calculators and virtual manipulatives that support. conceptual understanding and higher order thinking skills During classroom instruction students should have access to various mathematical. tools as well as paper and determine which tools are the most appropriate to use For example when comparing 4 6 and 1 2 students can use. benchmark fractions and the number line and explain that 4 6 would be placed to the right of because it is a little bit more than When. students model situations with mathematics they are choosing tools appropriately SMP 5 As they decontextualize the situation and represent. it mathematically they are also reasoning abstractly SMP2. 6 Attend to Mathematically proficient students in Grade 3 are precise in their communication calculations and measurements In all mathematical tasks. precision they communicate clearly using grade level appropriate vocabulary accurately as well as giving precise explanations and reasoning regarding. their process of finding solutions For example while measuring objects iteratively repetitively students check to make sure that there are no. gaps or overlaps In using representations such as pictures tables graphs or diagrams they use appropriate labels to communicate the meaning. of their representation During tasks involving number sense students check their work to ensure the accuracy and reasonableness of solutions. 7 Look for and Mathematically proficient students in Grade 3 carefully look for patterns and structures in the number system and other areas of mathematics. make use of Grade 3 students use structures such as place value the properties of operations other generalizations about the behavior of the operations for. structure example the less you subtract the greater the difference and attributes of shapes to solve problems In many cases they have identified and. described these structures through repeated reasoning SMP 8 For example when Grade 3 students calculate 16 X 9 they might apply the. structure of place value and the distributive property to find the product 16 X 9 10 6 X 9 10 X 9 6 X 9. 8 Look for and Mathematically proficient students in Grade 3 notice repetitive actions in computation and look for more shortcut methods For example. express students may use the distributive property as a strategy for using products they know to solve products that they don t know For example if. regularity in students are asked to find the product of 7 X 8 they might decompose 7 into 5 and 2 and then multiply 5 X 8 and 2 X 8 to arrive at 40 16 or 56. repeated Mathematically proficient 3rd graders formulate conjectures about what they notice In addition third graders continually evaluate their work by. reasoning asking themselves Does this make sense, Summary of Standards for Mathematical Practice Questions to Develop Mathematical Thinking. 1 Make sense of problems and persevere in solving them How would you describe the problem in your own words. Interpret and make meaning of the problem looking for How would you describe what you are trying to find. starting points Analyze what is given to explain to What do you notice about. themselves the meaning of the problem What information is given in the problem. Plan a solution pathway instead of jumping to a solution Describe the relationship between the quantities. Can monitor their progress and change the approach if Describe what you have already tried. necessary What might you change, See relationships between various representations Talk me through the steps you ve used to this point. Relate current situations to concepts or skills previously What steps in the process are you most confident about. learned and connect mathematical ideas to one another What are some other strategies you might try. Can understand various approaches to solutions What are some other problems that are similar to this one. Continually ask themselves Does this make sense How might you use one of your previous problems to help. How else might you organize represent and show, 2 Reason abstractly and quantitatively What do the numbers used in the problem represent. Make sense of quantities and their relationships What is the relationship of the quantities. Are able to decontextualize represent a situation How is related to. symbolically and manipulate the symbols and What is the relationship between and. contextualize make meaning of the symbols in a What does mean to you e g symbol quantity. problem quantitative relationships diagram, Understand the meaning of quantities and are flexible in What properties might we use to find a solution. the use of operations and their properties How did you decide in this task that you needed to use. Create a logical representation of the problem Could we have used another operation or property to solve. Attends to the meaning of quantities not just how to this task Why or why not. compute them, 3 Construct viable arguments and critique the reasoning of What mathematical evidence would support your solution. others How can we be sure that How could you prove. Analyze problems and use stated mathematical that Will it still work if. assumptions definitions and established results in What were you considering when. constructing arguments How did you decide to try that strategy. Justify conclusions with mathematical ideas How did you test whether your approach worked. Listen to the arguments of others and ask useful How did you decide what the problem was asking you to. questions to determine if an argument makes sense find What was unknown. Ask clarifying questions or suggest ideas to Did you try a method that did not work Why didn t it work. improve revise the argument Would it ever work Why or why not. Compare two arguments and determine correct or What is the same and what is different about. flawed logic How could you demonstrate a counter example. 4 Model with mathematics What number model could you construct to represent the. Understand this is a way to reason quantitatively and problem. abstractly able to decontextualize and contextualize What are some ways to represent the quantities. Apply the math they know to solve problems in everyday What s an equation or expression that matches the diagram. life number line chart table, Are able to simplify a complex problem and identify Where did you see one of the quantities in the task in your. important quantities to look at relationships equation or expression. Represent mathematics to describe a situation either Would it help to create a diagram graph table. with an equation or a diagram and interpret the results What are some ways to visually represent. of a mathematical situation What formula might apply in this situation. Reflect on whether the results make sense possibly. improving or revising the model,Ask themselves How can I represent this. mathematically, Summary of Standards for Mathematical Practice Questions to Develop Mathematical Thinking. 5 Use appropriate tools strategically What mathematical tools could we use to visualize and. Use available tools recognizing the strengths and represent the situation. limitations of each What information do you have, Use estimation and other mathematical knowledge to What do you know that is not stated in the problem. detect possible errors What approach are you considering trying first. Identify relevant external mathematical resources to What estimate did you make for the solution. pose and solve problems In this situation would it be helpful to use a graph number. Use technological tools to deepen their understanding of line ruler diagram calculator manipulative. mathematics Why was it helpful to use,What can using a show us that may not. In what situations might it be more informative or helpful to. 6 Attend to precision What mathematical terms apply in this situation. Communicate precisely with others and try to use clear How did you know your solution was reasonable. mathematical language when discussing their reasoning Explain how you might show that your solution answers the. Understand meanings of symbols used in mathematics problem. and can label quantities appropriately Is there a more efficient strategy. Express numerical answers with a degree of precision How are you showing the meaning of the quantities. appropriate for the problem context What symbols or mathematical notations are important in. Calculate efficiently and accurately this problem, What mathematical language definitions properties can. you use to explain, How could you test your solution to see if it answers the. 7 Look for and make use of structure What observations do you make about. Apply general mathematical rules to specific situations What do you notice when. Look for the overall structure and patterns in What parts of the problem might you eliminate simplify. mathematics What patterns do you find in, See complicated things as single objects or as being How do you know if something is a pattern. composed of several objects What ideas that we have learned before were useful in. solving this problem, What are some other problems that are similar to this one. How does this relate to,In what ways does this problem connect to other. mathematical concepts, 8 Look for and express regularity in repeated reasoning Will the same strategy work in other situations. See repeated calculations and look for generalizations Is this always true sometimes true or never true. and shortcuts How would we prove that, See the overall process of the problem and still attend to What do you notice about. the details What is happening in this situation, Understand the broader application of patterns and see What would happen if. the structure in similar situations What Is there a mathematical rule for. Continually evaluate the reasonableness of their What predictions or generalizations can this pattern. intermediate results support,What mathematical consistencies do you notice. Critical Areas for Mathematics in 3rd Grade, In Grade 3 instructional time should focus on four critical areas 1 developing understanding of. multiplication and division and strategies for multiplication and division within 100 2 developing. understanding of fractions especially unit fractions fractions with numerator 1 3 developing. understanding of the structure of rectangular arrays and of area and 4 describing and analyzing two. dimensional shapes, 1 Students develop an understanding of the meanings of multiplication and division of whole numbers. through activities and problems involving equal sized groups arrays and area models multiplication is finding. an unknown product and division is finding an unknown factor in these situations For equal sized group. situations division can require finding the unknown number of groups or the unknown group size Students. use properties of operations to calculate products of whole numbers using increasingly sophisticated. strategies based on these properties to solve multiplication and division problems involving single digit. factors By comparing a variety of solution strategies students learn the relationship between multiplication. and division OA 1 OA 2 OA 3 OA 4 OA 5 OA 6 OA 7 OA 9. 2 Students develop an understanding of fractions beginning with unit fractions Students view fractions in. general as being built out of unit fractions and they use fractions along with visual fraction models to. represent parts of a whole Students understand that the size of a fractional part is relative to the size of the. whole For example 1 2 of the paint in a small bucket could be less paint than 1 3 of the paint in a larger. bucket but 1 3 of a ribbon is longer than 1 5 of the same ribbon because when the ribbon is divided into 3. equal parts the parts are longer than when the ribbon is divided into 5 equal parts Students are able to use. fractions to represent numbers equal to less than and greater than one They solve problems that involve. comparing fractions by using visual fraction models and strategies based on noticing equal numerators or. denominators,NF 1 NF 2 NF 3, 3 Students recognize area as an attribute of two dimensional regions They measure the area of a shape by. finding the total number of same size units of area required to cover the shape without gaps or overlaps a. square with sides of unit length being the standard unit for measuring area Students understand that. rectangular arrays can be decomposed into identical rows or into identical columns By decomposing. rectangles into rectangular arrays of squares students connect area to multiplication and justify using. multiplication to determine the area of a rectangle. MD 5 MD 6 MD 7, 4 Students describe analyze and compare properties of two dimensional shapes They compare and classify. shapes by their sides and angles and connect these with definitions of shapes Students also relate their. fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole. NF 1 G 1 G 2,Dynamic Learning Maps DLM and Essential Elements. The Dynamic Learning Maps and Essential Elements are knowledge and skills linked to the grade level expectations. identified in the Common Core State Standards The purpose of the Dynamic Learning Maps Essential Elements is to. build a bridge from the content in the Common Core State Standards to academic expectations for students with the. most significant cognitive disabilities, For more information please visit the Dynamic Learning Maps and Essential Elements website. Grade 3 Content Standards Overview,Operations and Algebraic Thinking OA. Represents and solves problems involving multiplication and division. OA 1 OA 2 OA 3 OA 4, Understand properties of multiplication and the relationship between multiplication and division. Multiply and divide within 100, Solve problems involving the four operations and identify and explain patterns in arithmetic. Number and Operations in Base Ten NBT, Use place value understanding and properties of operations to perform multi digit arithmetic. NBT 1 NBT 2 NBT 3,Number and Operations Fractions NF. Develop understanding of fractions as numbers,NF 1 NF 2 NF 3. Measurement and Data MD, Solve problems involving measurement and estimation of intervals of time liquid volumes and masses of. Represent and interpret data, Geometric measurement understand concepts of area and relate area to multiplication and to addition. MD 5 MD 6 MD 7, Geometric measurement recognize perimeter as an attribute of plane figures and distinguish between linear. and area measures,Geometry GE,Reason with shapes and their attributes. Domain Operations and Algebraic Thinking OA, Cluster Represents and solves problems involving multiplication and division. Standard Grade 3 OA 1, Interpret products of whole numbers e g interpret 5 7 as the total number of objects in 5 groups. of 7 objects each For example describe a context in which a total number of objects can be. expressed as 5 7 OA,Suggested Standards for Mathematical Practice MP. MP 1 Make sense of problems and persevere in solving them. MP 2 Reason abstractly and quantitatively,MP 4 Model with mathematics. MP 6 Attend to precision,MP 7 Look for and make use of structure. Connections 3 OA 1 4, This cluster is connected to the Third Grade Critical Area of Focus 1. Developing understanding of multiplication and division and strategies for multiplication and. division within 100, Connect this domain with understanding properties of multiplication and the relationship between. multiplication and division Grade 3 OA 5 6, The use of a symbol for an unknown is foundational for letter variables in Grade 4 when. representing problems using equations with a letter standing for the unknown quantity Grade 4 OA. 2 and OA 3,Explanation and Examples, The standard interprets products of whole numbers Students need to recognize multiplication as a. means of determining the total number of objects when there are a specific number of groups with. the same number of objects in each group Multiplication requires students to think in terms of. groups of things rather than individual things At this level Multiplication is seen as groups of and. problems such as 5 x 7 refer to 5 groups of 7, It is important for teachers to understand there are several ways in which think of multiplication. Multiplication is often thought of as repeated addition of equal groups While this definition. works for some sets of numbers it is not particularly intuitive or meaningful when we think of. multiplying 3 by 1 2 for example or 5 by 2 In such cases it may be helpful to widen the idea. of grouping to include evaluation of part of a group This concept is related to partitioning. which in turn is related to division, Ex Three groups of five students can be read as 3 5 or 15 students while half a. group of 10 stars can be represented as 1 2 10 or 5 stars These are examples of. partitioning each one of the three groups of five is part of the group of 15 and the. group of 5 stars is part of the group of 10, A second concept of multiplication is that of rate or price Ex If a car travels four hours at 50. miles per hour then it travels a total of 4 50 or 200 miles if CDs cost eight dollars each then. three CDs will cost 3 8 or 24, A third concept of multiplication is that of multiplicative comparison Ex Sara has. four CDs Joanne has three times as many as Sara and Sylvia has half as many as Sara. Thus Joanne has 3 4 or 12 CDs and Sylvia has 1 2 4 or 2 CDs. Example for 3 OA 1, Jim purchased 5 packages of muffins Each package contained 3 muffins How many muffins did. Jim purchase 5 groups of 3 5 x 3 15, Describe another situation where there would be 5 groups of 3 or 5 x 3. Students recognize multiplication as a means to determine the total number of objects when there. are a specific number of groups with the same number of objects in each group Multiplication. requires students to think in terms of groups of things rather than individual things Students learn. that the multiplication symbol x means groups of and problems such as 5 x 7 refer to 5 groups of. To further develop this understanding students interpret a problem situation requiring. multiplication using pictures objects words numbers and equations Then given a multiplication. expression e g 5 x 6 students interpret the expression using a multiplication context See. Appendix for chart They should begin to use the terms factor and product as they describe.
Common Core State Standards For Mathematics Flip Book Grade 7 Updated Fall, 2014 This project used the work done by the Departments of Educations in Ohio, North Carolina, Georgia, engageNY, NCTM, and the Tools for the Common Core Standards. Compiled by Melisa J. Hancock, for questions or comments about the flipbooks please contact
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