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common core state standards for mathematics

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common core state standards for mathematics table of contents introduction standards for mathematical practice standards for mathematical content kindergarten grade 1 grade 2 grade 3 grade 4 grade 5 grade 6 grade 7 grade 8 high school introduction high school number and quantity high school algebra high school functions high school modeling high school geometry high school statistics and probability 3 6 9 13 17 21 27 33 39 46 52 58 62 67 72 74 79 85 91 glossary sample of works consulted

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common core state standards for mathematics introduction toward greater focus and coherence mathematics experiences in early childhood settings should concentrate on 1 number which includes whole number operations and relations and 2 geometry spatial relations and measurement with more mathematics learning time devoted to number than to other topics mathematical process goals should be integrated in these content areas mathematics learning in early childhood national research council 2009 the composite standards [of hong kong korea and singapore have a number of features that can inform an international benchmarking process for the development of k6 mathematics standards in the u.s first the composite standards concentrate the early learning of mathematics on the number measurement and geometry strands with less emphasis on data analysis and little exposure to algebra the hong kong standards for grades 13 devote approximately half the targeted time to numbers and almost all the time remaining to geometry and measurement ginsburg leinwand and decker 2009 because the mathematics concepts in [u.s textbooks are often weak the presentation becomes more mechanical than is ideal we looked at both traditional and non-traditional textbooks used in the us and found this conceptual weakness in both ginsburg et al 2005 there are many ways to organize curricula the challenge now rarely met is to avoid those that distort mathematics and turn off students steen 2007 for over a decade research studies of mathematics education in high-performing countries have pointed to the conclusion that the mathematics curriculum in the united states must become substantially more focused and coherent in order to improve mathematics achievement in this country to deliver on the promise of common standards the standards must address the problem of a curriculum that is a mile wide and an inch deep these standards are a substantial answer to that challenge it is important to recognize that fewer standards are no substitute for focused standards achieving fewer standards would be easy to do by resorting to broad general statements instead these standards aim for clarity and specificity assessing the coherence of a set of standards is more difficult than assessing their focus william schmidt and richard houang 2002 have said that content standards and curricula are coherent if they are articulated over time as a sequence of topics and performances that are logical and reflect where appropriate the sequential or hierarchical nature of the disciplinary content from which the subject matter derives that is what and how students are taught should reflect not only the topics that fall within a certain academic discipline but also the key ideas that determine how knowledge is organized and generated within that discipline this implies introduction 3

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common core state standards for mathematics that to be coherent a set of content standards must evolve from particulars e.g the meaning and operations of whole numbers including simple math facts and routine computational procedures associated with whole numbers and fractions to deeper structures inherent in the discipline these deeper structures then serve as a means for connecting the particulars such as an understanding of the rational number system and its properties emphasis added these standards endeavor to follow such a design not only by stressing conceptual understanding of key ideas but also by continually returning to organizing principles such as place value or the properties of operations to structure those ideas in addition the sequence of topics and performances that is outlined in a body of mathematics standards must also respect what is known about how students learn as confrey 2007 points out developing sequenced obstacles and challenges for students absent the insights about meaning that derive from careful study of learning would be unfortunate and unwise in recognition of this the development of these standards began with research-based learning progressions detailing what is known today about how students mathematical knowledge skill and understanding develop over time understanding mathematics these standards define what students should understand and be able to do in their study of mathematics asking a student to understand something means asking a teacher to assess whether the student has understood it but what does mathematical understanding look like one hallmark of mathematical understanding is the ability to justify in a way appropriate to the student s mathematical maturity why a particular mathematical statement is true or where a mathematical rule comes from there is a world of difference between a student who can summon a mnemonic device to expand a product such as a b x y and a student who can explain where the mnemonic comes from the student who can explain the rule understands the mathematics and may have a better chance to succeed at a less familiar task such as expanding a b c x y mathematical understanding and procedural skill are equally important and both are assessable using mathematical tasks of sufficient richness the standards set grade-specific standards but do not define the intervention methods or materials necessary to support students who are well below or well above grade-level expectations it is also beyond the scope of the standards to define the full range of supports appropriate for english language learners and for students with special needs at the same time all students must have the opportunity to learn and meet the same high standards if they are to access the knowledge and skills necessary in their post-school lives the standards should be read as allowing for the widest possible range of students to participate fully from the outset along with appropriate accommodations to ensure maximum participaton of students with special education needs for example for students with disabilities reading should allow for use of braille screen reader technology or other assistive devices while writing should include the use of a scribe computer or speech-to-text technology in a similar vein speaking and listening should be interpreted broadly to include sign language no set of grade-specific standards can fully reflect the great variety in abilities needs learning rates and achievement levels of students in any given classroom however the standards do provide clear signposts along the way to the goal of college and career readiness for all students the standards begin on page 6 with eight standards for mathematical practice introduction 4

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common core state standards for mathematics how to read the grade level standards standards define what students should understand and be able to do clusters are groups of related standards note that standards from different clusters may sometimes be closely related because mathematics is a connected subject domains are larger groups of related standards standards from different domains may sometimes be closely related domain number and operations in base ten 3.nbt use place value understanding and properties of operations to perform multi-digit arithmetic 1 standard use place value understanding to round whole numbers to the nearest 10 or 100 cluster 2 fluently add and subtract within 1000 using strategies and algorithms based on place value properties of operations and/or the relationship between addition and subtraction 3 multiply one-digit whole numbers by multiples of 10 in the range 1090 e.g 9 × 80 5 × 60 using strategies based on place value and properties of operations these standards do not dictate curriculum or teaching methods for example just because topic a appears before topic b in the standards for a given grade it does not necessarily mean that topic a must be taught before topic b a teacher might prefer to teach topic b before topic a or might choose to highlight connections by teaching topic a and topic b at the same time or a teacher might prefer to teach a topic of his or her own choosing that leads as a byproduct to students reaching the standards for topics a and b what students can learn at any particular grade level depends upon what they have learned before ideally then each standard in this document might have been phrased in the form students who already know should next come to learn but at present this approach is unrealistic not least because existing education research cannot specify all such learning pathways of necessity therefore grade placements for specific topics have been made on the basis of state and international comparisons and the collective experience and collective professional judgment of educators researchers and mathematicians one promise of common state standards is that over time they will allow research on learning progressions to inform and improve the design of standards to a much greater extent than is possible today learning opportunities will continue to vary across schools and school systems and educators should make every effort to meet the needs of individual students based on their current understanding these standards are not intended to be new names for old ways of doing business they are a call to take the next step it is time for states to work together to build on lessons learned from two decades of standards based reforms it is time to recognize that standards are not just promises to our children but promises we intend to keep introduction 5

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common core state standards for mathematics mathematics standards for mathematical practice the standards for mathematical practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students these practices rest on important processes and proficiencies with longstanding importance in mathematics education the first of these are the nctm process standards of problem solving reasoning and proof communication representation and connections the second are the strands of mathematical proficiency specified in the national research council s report adding it up adaptive reasoning strategic competence conceptual understanding comprehension of mathematical concepts operations and relations procedural fluency skill in carrying out procedures flexibly accurately efficiently and appropriately and productive disposition habitual inclination to see mathematics as sensible useful and worthwhile coupled with a belief in diligence and one s own efficacy 1 make sense of problems and persevere in solving them mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution they analyze givens constraints relationships and goals they make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt they consider analogous problems and try special cases and simpler forms of the original problem in order to gain insight into its solution they monitor and evaluate their progress and change course if necessary older students might depending on the context of the problem transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need mathematically proficient students can explain correspondences between equations verbal descriptions tables and graphs or draw diagrams of important features and relationships graph data and search for regularity or trends younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem mathematically proficient students check their answers to problems using a different method and they continually ask themselves does this make sense they can understand the approaches of others to solving complex problems and identify correspondences between different approaches 2 reason abstractly and quantitatively mathematically proficient students make sense of quantities and their relationships in problem situations they bring two complementary abilities to bear on problems involving quantitative relationships the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own without necessarily attending to their referents and the ability to contextualize to pause as needed during the manipulation process in order to probe into the referents for the symbols involved quantitative reasoning entails habits of creating a coherent representation of the problem at hand considering the units involved attending to the meaning of quantities not just how to compute them and knowing and flexibly using different properties of operations and objects standards for mathematical practice 3 construct viable arguments and critique the reasoning of others mathematically proficient students understand and use stated assumptions definitions and previously established results in constructing arguments they make conjectures and build a logical progression of statements to explore the truth of their conjectures they are able to analyze situations by breaking them into cases and can recognize and use counterexamples they justify their conclusions 6

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common core state standards for mathematics communicate them to others and respond to the arguments of others they reason inductively about data making plausible arguments that take into account the context from which the data arose mathematically proficient students are also able to compare the effectiveness of two plausible arguments distinguish correct logic or reasoning from that which is flawed and if there is a flaw in an argument explain what it is elementary students can construct arguments using concrete referents such as objects drawings diagrams and actions such arguments can make sense and be correct even though they are not generalized or made formal until later grades later students learn to determine domains to which an argument applies students at all grades can listen or read the arguments of others decide whether they make sense and ask useful questions to clarify or improve the arguments 4 model with mathematics mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life society and the workplace in early grades this might be as simple as writing an addition equation to describe a situation in middle grades a student might apply proportional reasoning to plan a school event or analyze a problem in the community by high school a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation realizing that these may need revision later they are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams two-way tables graphs flowcharts and formulas they can analyze those relationships mathematically to draw conclusions they routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense possibly improving the model if it has not served its purpose 5 use appropriate tools strategically mathematically proficient students consider the available tools when solving a mathematical problem these tools might include pencil and paper concrete models a ruler a protractor a calculator a spreadsheet a computer algebra system a statistical package or dynamic geometry software proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful recognizing both the insight to be gained and their limitations for example mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator they detect possible errors by strategically using estimation and other mathematical knowledge when making mathematical models they know that technology can enable them to visualize the results of varying assumptions explore consequences and compare predictions with data mathematically proficient students at various grade levels are able to identify relevant external mathematical resources such as digital content located on a website and use them to pose or solve problems they are able to use technological tools to explore and deepen their understanding of concepts standards for mathematical practice 6 attend to precision mathematically proficient students try to communicate precisely to others they try to use clear definitions in discussion with others and in their own reasoning they state the meaning of the symbols they choose including using the equal sign consistently and appropriately they are careful about specifying units of measure and labeling axes to clarify the correspondence with quantities in a problem they calculate accurately and efficiently express numerical answers with a degree of precision appropriate for the problem context in the elementary grades students give carefully formulated explanations to each other by the time they reach high school they have learned to examine claims and make explicit use of definitions 7

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common core state standards for mathematics 7 look for and make use of structure mathematically proficient students look closely to discern a pattern or structure young students for example might notice that three and seven more is the same amount as seven and three more or they may sort a collection of shapes according to how many sides the shapes have later students will see 7 × 8 equals the well remembered 7 × 5 7 × 3 in preparation for learning about the distributive property in the expression x2 9x 14 older students can see the 14 as 2 × 7 and the 9 as 2 7 they recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems they also can step back for an overview and shift perspective they can see complicated things such as some algebraic expressions as single objects or as being composed of several objects for example they can see 5 3x y2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y 8 look for and express regularity in repeated reasoning mathematically proficient students notice if calculations are repeated and look both for general methods and for shortcuts upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again and conclude they have a repeating decimal by paying attention to the calculation of slope as they repeatedly check whether points are on the line through 1 2 with slope 3 middle school students might abstract the equation y 2 x 1 3 noticing the regularity in the way terms cancel when expanding x 1 x 1 x 1 x2 x 1 and x 1 x3 x2 x 1 might lead them to the general formula for the sum of a geometric series as they work to solve a problem mathematically proficient students maintain oversight of the process while attending to the details they continually evaluate the reasonableness of their intermediate results connecting the standards for mathematical practice to the standards for mathematical content the standards for mathematical practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary middle and high school years designers of curricula assessments and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction the standards for mathematical content are a balanced combination of procedure and understanding expectations that begin with the word understand are often especially good opportunities to connect the practices to the content students who lack understanding of a topic may rely on procedures too heavily without a flexible base from which to work they may be less likely to consider analogous problems represent problems coherently justify conclusions apply the mathematics to practical situations use technology mindfully to work with the mathematics explain the mathematics accurately to other students step back for an overview or deviate from a known procedure to find a shortcut in short a lack of understanding effectively prevents a student from engaging in the mathematical practices in this respect those content standards which set an expectation of understanding are potential points of intersection between the standards for mathematical content and the standards for mathematical practice these points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time resources innovative energies and focus necessary to qualitatively improve the curriculum instruction assessment professional development and student achievement in mathematics standards for mathematical practice 8

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common core state standards for mathematics mathematics kindergarten in kindergarten instructional time should focus on two critical areas 1 representing relating and operating on whole numbers initially with sets of objects 2 describing shapes and space more learning time in kindergarten should be devoted to number than to other topics 1 students use numbers including written numerals to represent quantities and to solve quantitative problems such as counting objects in a set counting out a given number of objects comparing sets or numerals and modeling simple joining and separating situations with sets of objects or eventually with equations such as 5 2 7 and 7 2 5 kindergarten students should see addition and subtraction equations and student writing of equations in kindergarten is encouraged but it is not required students choose combine and apply effective strategies for answering quantitative questions including quickly recognizing the cardinalities of small sets of objects counting and producing sets of given sizes counting the number of objects in combined sets or counting the number of objects that remain in a set after some are taken away 2 students describe their physical world using geometric ideas e.g shape orientation spatial relations and vocabulary they identify name and describe basic two-dimensional shapes such as squares triangles circles rectangles and hexagons presented in a variety of ways e.g with different sizes and orientations as well as three-dimensional shapes such as cubes cones cylinders and spheres they use basic shapes and spatial reasoning to model objects in their environment and to construct more complex shapes kindergarten 9

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common core state standards for mathematics grade k overview counting and cardinality · know number names and the count sequence · count to tell the number of objects · compare numbers mathematical practices 1 2 3 4 5 6 7 number and operations in base ten · work with numbers 1119 to gain foundations for place value 8 make sense of problems and persevere in solving them reason abstractly and quantitatively construct viable arguments and critique the reasoning of others model with mathematics use appropriate tools strategically attend to precision look for and make use of structure look for and express regularity in repeated reasoning operations and algebraic thinking · understand addition as putting together and adding to and understand subtraction as taking apart and taking from measurement and data · describe and compare measurable attributes · classify objects and count the number of objects in categories geometry · identify and describe shapes · analyze compare create and compose shapes kindergarten 10

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common core state standards for mathematics counting and cardinality know number names and the count sequence 1 count to 100 by ones and by tens 2 count forward beginning from a given number within the known sequence instead of having to begin at 1 3 write numbers from 0 to 20 represent a number of objects with a written numeral 0-20 with 0 representing a count of no objects k.cc count to tell the number of objects 4 understand the relationship between numbers and quantities connect counting to cardinality a b c when counting objects say the number names in the standard order pairing each object with one and only one number name and each number name with one and only one object understand that the last number name said tells the number of objects counted the number of objects is the same regardless of their arrangement or the order in which they were counted understand that each successive number name refers to a quantity that is one larger 5 count to answer how many questions about as many as 20 things arranged in a line a rectangular array or a circle or as many as 10 things in a scattered configuration given a number from 120 count out that many objects compare numbers 6 identify whether the number of objects in one group is greater than less than or equal to the number of objects in another group e.g by using matching and counting strategies.1 7 compare two numbers between 1 and 10 presented as written numerals operations and algebraic thinking k.oa understand addition as putting together and adding to and understand subtraction as taking apart and taking from 1 represent addition and subtraction with objects fingers mental images drawings2 sounds e.g claps acting out situations verbal explanations expressions or equations 2 solve addition and subtraction word problems and add and subtract within 10 e.g by using objects or drawings to represent the problem 3 decompose numbers less than or equal to 10 into pairs in more than one way e.g by using objects or drawings and record each decomposition by a drawing or equation e.g 5 2 3 and 5 4 1 4 for any number from 1 to 9 find the number that makes 10 when added to the given number e.g by using objects or drawings and record the answer with a drawing or equation 5 fluently add and subtract within 5 1 2 include groups with up to ten objects drawings need not show details but should show the mathematics in the problem this applies wherever drawings are mentioned in the standards kindergarten 11

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common core state standards for mathematics number and operations in base ten work with numbers 1119 to gain foundations for place value 1 k.nbt compose and decompose numbers from 11 to 19 into ten ones and some further ones e.g by using objects or drawings and record each composition or decomposition by a drawing or equation e.g 18 10 8 understand that these numbers are composed of ten ones and one two three four five six seven eight or nine ones measurement and data describe and compare measurable attributes 1 k.md describe measurable attributes of objects such as length or weight describe several measurable attributes of a single object 2 directly compare two objects with a measurable attribute in common to see which object has more of less of the attribute and describe the difference for example directly compare the heights of two children and describe one child as taller/shorter classify objects and count the number of objects in each category 3 classify objects into given categories count the numbers of objects in each category and sort the categories by count.3 geometry k.g identify and describe shapes squares circles triangles rectangles hexagons cubes cones cylinders and spheres 1 describe objects in the environment using names of shapes and describe the relative positions of these objects using terms such as above below beside in front of behind and next to 2 correctly name shapes regardless of their orientations or overall size 3 identify shapes as two-dimensional lying in a plane flat or threedimensional solid analyze compare create and compose shapes 4 analyze and compare two and three-dimensional shapes in different sizes and orientations using informal language to describe their similarities differences parts e.g number of sides and vertices corners and other attributes e.g having sides of equal length 5 model shapes in the world by building shapes from components e.g sticks and clay balls and drawing shapes 6 compose simple shapes to form larger shapes for example can you join these two triangles with full sides touching to make a rectangle kindergarten 3 limit category counts to be less than or equal to 10 12

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common core state standards for mathematics mathematics grade 1 in grade 1 instructional time should focus on four critical areas 1 developing understanding of addition subtraction and strategies for addition and subtraction within 20 2 developing understanding of whole number relationships and place value including grouping in tens and ones 3 developing understanding of linear measurement and measuring lengths as iterating length units and 4 reasoning about attributes of and composing and decomposing geometric shapes 1 students develop strategies for adding and subtracting whole numbers based on their prior work with small numbers they use a variety of models including discrete objects and length-based models e.g cubes connected to form lengths to model add-to take-from put-together take-apart and compare situations to develop meaning for the operations of addition and subtraction and to develop strategies to solve arithmetic problems with these operations students understand connections between counting and addition and subtraction e.g adding two is the same as counting on two they use properties of addition to add whole numbers and to create and use increasingly sophisticated strategies based on these properties e.g making tens to solve addition and subtraction problems within 20 by comparing a variety of solution strategies children build their understanding of the relationship between addition and subtraction 2 students develop discuss and use efficient accurate and generalizable methods to add within 100 and subtract multiples of 10 they compare whole numbers at least to 100 to develop understanding of and solve problems involving their relative sizes they think of whole numbers between 10 and 100 in terms of tens and ones especially recognizing the numbers 11 to 19 as composed of a ten and some ones through activities that build number sense they understand the order of the counting numbers and their relative magnitudes 3 students develop an understanding of the meaning and processes of measurement including underlying concepts such as iterating the mental activity of building up the length of an object with equal-sized units and the transitivity principle for indirect measurement.1 4 students compose and decompose plane or solid figures e.g put two triangles together to make a quadrilateral and build understanding of part-whole relationships as well as the properties of the original and composite shapes as they combine shapes they recognize them from different perspectives and orientations describe their geometric attributes and determine how they are alike and different to develop the background for measurement and for initial understandings of properties such as congruence and symmetry students should apply the principle of transitivity of measurement to make indirect comparisons but they need not use this technical term 1 grade 1 13

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common core state standards for mathematics grade 1 overview operations and algebraic thinking · represent and solve problems involving addition and subtraction · understand and apply properties of operations and the relationship between addition and subtraction · add and subtract within 20 · work with addition and subtraction equations mathematical practices 1 2 3 4 5 6 number and operations in base ten · extend the counting sequence · understand place value · use place value understanding and properties of operations to add and subtract 7 8 make sense of problems and persevere in solving them reason abstractly and quantitatively construct viable arguments and critique the reasoning of others model with mathematics use appropriate tools strategically attend to precision look for and make use of structure look for and express regularity in repeated reasoning measurement and data · measure lengths indirectly and by iterating length units · tell and write time · represent and interpret data geometry · reason with shapes and their attributes grade 1 14

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common core state standards for mathematics operations and algebraic thinking 1 1.oa represent and solve problems involving addition and subtraction use addition and subtraction within 20 to solve word problems involving situations of adding to taking from putting together taking apart and comparing with unknowns in all positions e.g by using objects drawings and equations with a symbol for the unknown number to represent the problem.2 2 solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20 e.g by using objects drawings and equations with a symbol for the unknown number to represent the problem understand and apply properties of operations and the relationship between addition and subtraction 3 apply properties of operations as strategies to add and subtract.3 examples if 8 3 11 is known then 3 8 11 is also known commutative property of addition to add 2 6 4 the second two numbers can be added to make a ten so 2 6 4 2 10 12 associative property of addition 4 understand subtraction as an unknown-addend problem for example subtract 10 8 by finding the number that makes 10 when added to 8 add and subtract within 20 5 relate counting to addition and subtraction e.g by counting on 2 to add 2 6 add and subtract within 20 demonstrating fluency for addition and subtraction within 10 use strategies such as counting on making ten e.g 8 6 8 2 4 10 4 14 decomposing a number leading to a ten e.g 13 4 13 3 1 10 1 9 using the relationship between addition and subtraction e.g knowing that 8 4 12 one knows 12 8 4 and creating equivalent but easier or known sums e.g adding 6 7 by creating the known equivalent 6 6 1 12 1 13 work with addition and subtraction equations 7 understand the meaning of the equal sign and determine if equations involving addition and subtraction are true or false for example which of the following equations are true and which are false 6 6 7 8 1 5 2 2 5 4 1 5 2 8 determine the unknown whole number in an addition or subtraction equation relating three whole numbers for example determine the unknown number that makes the equation true in each of the equations 8 11 5 3 6 6 number and operations in base ten extend the counting sequence 1 1.nbt count to 120 starting at any number less than 120 in this range read and write numerals and represent a number of objects with a written numeral understand place value 2 understand that the two digits of a two-digit number represent amounts of tens and ones understand the following as special cases a b c 2 10 can be thought of as a bundle of ten ones called a ten the numbers from 11 to 19 are composed of a ten and one two three four five six seven eight or nine ones the numbers 10 20 30 40 50 60 70 80 90 refer to one two three four five six seven eight or nine tens and 0 ones grade 1 3 see glossary table 1 students need not use formal terms for these properties 15

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