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- 1. K Grade Mathematics ● Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13. This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers.What is the purpose of this document?To increase student achievement by ensuring educators understand specifically what the new standards mean a student must know,understand and be able to do.What is in the document?Descriptions of what each standard means a student will know, understand and be able to do. The “unpacking” of the standards done in thisdocument is an effort to answer a simple question “What does this standard mean that a student must know and be able to do?” and toensure the description is helpful, specific and comprehensive for educators.How do I send Feedback?We intend the explanations and examples in this document to be helpful and specific. That said, we believe that as this document is used,teachers and educators will find ways in which the unpacking can be improved and made ever more useful. Please send feedback to us atfeedback@dpi.state.nc.us and we will use your input to refine our unpacking of the standards. Thank You!Just want the standards alone?You can find the standards alone at http://corestandards.org/the-standardsMathematical Vocabulary is identified in bold print. These are words that students should know and be able to use in context. Kindergarten Mathematics ● Unpacked Content
- 2. Critical Areas and Changes in KindergartenCritical Areas: 1. Developing concepts of counting & cardinality 2. Developing beginning understanding of operations & algebraic thinking 3. Identifying and describing shapes in spaceNew to Kindergarten: Fluently add and subtract within 5 (K.CC.5) Compose and decompose numbers from 11 to 19 into tens and ones (K.NBT.1) Non-specification of shapes (K.G) Identify shapes as two-dimensional or three-dimensional (K.G.3) Compose simple shapes to form larger shapes (K.G.6)Moved from Kindergarten: Equal Shares (1.02) Calendar & Time (2.02) Data Collection (4.01, 4.02) Repeating Patterns (5.02)Note:Topics may appear to be similar between the CCSS and the 2003 NCSCOS; however, theCCSS may be presented at a higher cognitive demand.Kindergarten Mathematics ● Unpacked Content
- 3. Standards for Mathematical Practice in KindergartenThe Common Core State Standards for Mathematical Practice are practices expected to be integrated into every mathematics lesson for all studentsGrades K-12. Below are a few examples of how these Practices may be integrated into tasks that Kindergarten students complete.Practice Explanation and Example 1) Make Sense Mathematically proficient students in Kindergarten examine problems (tasks), can make sense of the meaning of the task and and Persevere find an entry point or a way to start the task. Kindergarten students also begin to develop a foundation for problem solving in Solving strategies and become independently proficient on using those strategies to solve new tasks. In Kindergarten, students‟ work Problems. focuses on concrete manipulatives before moving to pictorial representations. Kindergarten students also 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. Lastly, at the end of a task mathematically proficient students in Kindergarten ask themselves the question, “Does my answer make sense?” 2) Reason Mathematically proficient students in Kindergarten make sense of quantities and the relationships while solving tasks. This abstractly and involves two processes- decontexualizing and contextualizing. In Kindergarten, students represent situations by quantitatively. decontextualizing tasks into numbers and symbols. For example, in the task, “There are 7 children on the playground and some children go line up. If there are 4 children still playing, how many children lined up?” Kindergarten students are expected to translate that situation into the equation: 7-4 = ___, and then solve the task. Students also contextualize situations during the problem solving process. For example, while solving the task above, students refer to the context of the task to determine that they need to subtract 4 since the number of children on the playground is the total number of students except for the 4 that are still playing. Abstract reasoning also occurs when students measure and compare the lengths of objects. 3) Construct Mathematically proficient students in Kindergarten accurately use mathematical terms to construct arguments and engage in viable discussions about problem solving strategies. For example, while solving the task, “There are 8 books on the shelf. If you take arguments and some books off the shelf and there are now 3 left, how many books did you take off the shelf?” students will solve the task, and critique the then be able to construct an accurate argument about why they subtracted 3 form 8 rather than adding 8 and 3. Further, reasoning of Kindergarten students are expected to examine a variety of problem solving strategies and begin to recognize the others. reasonableness of them, as well as similarities and differences among them. 4) Model with Mathematically proficient students in Kindergarten model real-life mathematical situations with a number sentence or an mathematics. equation, and check to make sure that their equation accurately matches the problem context. Kindergarten students rely on concrete manipulatives and pictorial representations while solving tasks, but the expectation is that they will also write an equation to model problem situations. For example, while solving the task “there are 7 bananas on the counter. If you eat 3 bananas, how many are left?” Kindergarten students are expected to write the equation 7-3 = 4. Likewise, Kindergarten students are expected to create an appropriate problem situation from an equation. For example, students are expected to orally tell a story problem for the equation 4+5 = 9.Kindergarten Mathematics ● Unpacked Content
- 4. 5) Use Mathematically proficient students in Kindergarten have access to and use tools appropriately. These tools may include appropriate counters, place value (base ten) blocks, hundreds number boards, number lines, and concrete geometric shapes (e.g., pattern tools blocks, 3-d solids). Students should also have experiences with educational technologies, such as calculators, virtual strategically. manipulatives, and mathematical games that support conceptual understanding. 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, while solving the task “There are 4 dogs in the park. If 3 more dogs show up, how many dogs are they?” Kindergarten students are expected to explain why they used specific mathematical tools.” 6) Attend to Mathematically proficient students in Kindergarten are precise in their communication, calculations, and measurements. In allprecision mathematical tasks, students in Kindergarten describe their actions and strategies 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. 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 Kindergarten carefully look for patterns and structures in the number system and othermake use of areas of mathematics. While solving addition problems, students begin to recognize the commutative property, in that 1+4 = 5,structure and 4+1 = 5. While decomposing teen numbers, students realize that every number between 11 and 19, can be decomposed into 10 and some leftovers, such as 12 = 10+2, 13 = 10+3, etc. Further, Kindergarten students make use of structures of mathematical tasks when they begin to work with subtraction as missing addend problems, such as 5- 1 = __ can be written as 1+ __ = 5 and can be thought of as how much more do I need to add to 1 to get to 5?8) Look for and Mathematically proficient students in Kindergarten begin to look for regularity in problem structures when solvingexpress mathematical tasks. Likewise, students begin composing and decomposing numbers in different ways. For example, in the taskregularity in “There are 8 crayons in the box. Some are red and some are blue. How many of each could there be?” Kindergarten studentsrepeated are expected to realize that the 8 crayons could include 4 of each color (4+4 = 8), 5 of one color and 3 of another (5+3 = 8), etc.reasoning. For each solution, students repeated engage in the process of finding two numbers that can be joined to equal 8.Kindergarten Mathematics ● Unpacked Content page 4
- 5. Kindergarten Critical Areas(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, oreventually with equations such as 5 + 2 = 7 and 7 – 2 = 5. (Kindergarten students should see addition and subtraction equations, and studentwriting of equations in kindergarten is encouraged, but it is not required.) Students choose, combine, and apply effective strategies for answeringquantitative questions, including quickly recognizing the cardinalities ofsmall sets of objects, counting and producing sets of given sizes, counting the number of objects in combined sets, or counting the number ofobjects 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., withdifferent sizes and orientations), as well as three-dimensional shapes such as cubes, cones, cylinders, and spheres. They use basic shapes andspatial reasoning to model objects in their environment and to construct more complex shapes.Kindergarten Mathematics ● Unpacked Content page 5
- 6. Counting and Cardinality K.CCCommon Core Standard and ClusterKnow number names and the count sequence. Common Core Standard Unpacking What do these standards mean a child will know and be able to do?K.CC.1 Count to 100 by ones and by K.CC.1 calls for students to rote count by starting at one and count to 100. When students count by tens they aretens. only expected to master counting on the decade (0, 10, 20, 30, 40 …). This objective does not require recognition of numerals. It is focused on the rote number sequence.K.CC.2 Count forward beginning from K.CC.2 includes numbers 0 to 100. This asks for students to begin a rote forward counting sequence from aa given number within the known number other than 1. Thus, given the number 4, the student would count, “4, 5, 6 …” This objective does notsequence (instead of having to begin at require recognition of numerals. It is focused on the rote number sequence.1).K.CC.3 Write numbers from 0 to 20. K.CC.3 addresses the writing of numbers and using the written numerals (0-20) to describe the amount of a set ofRepresent a number of objects with a objects. Due to varied development of fine motor and visual development, a reversal of numerals is anticipatedwritten numeral 0-20 (with 0 for a majority of the students. While reversals should be pointed out to students, the emphasis is on the use ofrepresenting a count of no objects). numerals to represent quantities rather than the correct handwriting formation of the actual numeral itself. K.CC.3 asks for students to represent a set of objects with a written numeral. The number of objects being recorded should not be greater than 20. Students can record the quantity of a set by selecting a number card/tile (numeral recognition) or writing the numeral. Students can also create a set of objects based on the numeral presented.Common Core ClusterCount to tell the number of objects.Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such as counting objects in a set; counting outa given number of objects and comparing sets or numerals.Common Core Standard Unpacking What do these standards mean a child will know and be able to do?K.CC.4 Understand the relationship K.CC.4 asks students to count a set of objects and see sets and numerals in relationship to one another, ratherbetween numbers and quantities; than as isolated numbers or sets. These connections are higher-level skills that require students to analyze, toconnect counting to cardinality. reason about, and to explain relationships between numbers and sets of objects. This standard should first be addressed using numbers 1-5 with teachers building to the numbers 1-10 later in the year. The expectation is that Kindergarten Mathematics ● Unpacked Content page 6
- 7. students are comfortable with these skills with the numbers 1-10 by the end of Kindergarten.a. When counting objects, say the K.CC.4a reflects the ideas that students implement correct counting procedures by pointing to one object at anumber names in the standard order, time (one-to-one correspondence) using one counting word for every object (one-to-one tagging/synchrony),pairing each object with one and only while keeping track of objects that have and have not been counted.. This is the foundation of counting.one number name and each numbername with one and only one object.b. Understand that the last number K.CC.4b calls for students to answer the question “How many are there?” by counting objects in a set andname said tells the number of objects understanding that the last number stated when counting a set (…8, 9, 10) represents the total amount of objects:counted. The number of objects is the “There are 10 bears in this pile.” (cardinality). It also requires students to understand that the same set countedsame regardless of their arrangement or three different times will end up being the same amount each time. Thus, a purpose of keeping track of objects isthe order in which they were counted. developed. Therefore, a student who moves each object as it is counted recognizes that there is a need to keep track in order to figure out the amount of objects present. While it appears that this standard calls for students to have conservation of number, (regardless of the arrangement of objects, the quantity remains the same), conservation of number is a developmental milestone of which some Kindergarten children will not have mastered. The goal of this objective is for students to be able to count a set of objects; regardless of the formation those objects are placed.c. Understand that each successive K.CC.4c represents the concept of “one more” while counting a set of objects. Students are to make thenumber name refers to a quantity that is connection that if a set of objects was increased by one more object then the number name for that set is to beone larger. increased by one as well. Students are asked to understand this concept with and without objects. For example, after counting a set of 8 objects, students should be able to answer the question, “How many would there be if we added one more object?”; and answer a similar question when not using objects, by asking hypothetically, “What if we have 5 cubes and added one more. How many cubes would there be then?” This concept should be first taught with numbers 1-5 before building to numbers 1-10. Students should be expected to be comfortable with this skill with numbers to 10 by the end of Kindergarten.K.CC.5 Count to answer “how many?” K.CC.5 addresses various counting strategies. Based on early childhood mathematics experts, such as Kathyquestions about as many as 20 things Richardson, students go through a progression of four general ways to count. These counting strategies progressarranged in a line, a rectangular array, from least difficult to most difficult. First, students move objects and count them as they move them. The secondor a circle, or as many as 10 things in a strategy is that students line up the objects and count them. Third, students have a scattered arrangement and theyscattered configuration; given a number touch each object as they count. Lastly, students have a scattered arrangement and count them by visuallyfrom 1–20, count out that many objects. scanning without touching them. Since the scattered arrangements are the most challenging for students, K.CC.5 calls for students to only count 10 objects in a scattered arrangement, and count up to 20 objects in a line, rectangular array, or circle. Out of these 3 representations, a line is the easiest type of arrangement to count. Kindergarten Mathematics ● Unpacked Content page 7
- 8. Common Core ClusterCompare numbers. Common Core Standard Unpacking What do these standards mean a child will know and be able to do?K.CC.6 Identify whether the number of K.CC.6 expects mastery of up to ten objects. Students can use matching strategies (Student 1), countingobjects in one group is greater than, less strategies or equal shares (Student 3) to determine whether one group is greater than, less than, or equal to thethan, or equal to the number of objects number of objects in another group (Student 2).in another group, e.g., by usingmatching and counting strategies.1 Student 1 Student 2 Student 3 I lined up one square and one I counted the squares and I I put them in a pile. I then took1 Include groups with up to ten objects. triangle. Since there is one extra got 8. Then I counted the away objects. Every time I triangle, there are more triangles triangles and got 9. Since 9 is took a square, I also took a than squares. bigger than 8, there are more triangle. When I had taken triangles than squares. almost all of the shapes away, there was still a triangle left. That means that there are more triangles than squares.K.CC.7 Compare two numbers between K.CC.7 calls for students to apply their understanding of numerals 1-10 to compare one from another. Thus,1 and 10 presented as written numerals. looking at the numerals 8 and 10, a student must be able to recognize that the numeral 10 represents a larger amount than the numeral 8. Students should begin this standard by having ample experiences with sets of objects (K.CC.3 and K.CC.6) before completing this standard with just numerals. Based on early childhood research, students should not be expected to be comfortable with this skill until the end of Kindergarten. Kindergarten Mathematics ● Unpacked Content page 8
- 9. Operations and Algebraic Thinking K.0ACommon Core Standard and ClusterUnderstand addition as putting together and adding to, and understand subtraction as taking apart and taking from.All standards in this cluster should only include numbers through 10Students will model simple joining and separating situations with sets of objects, or eventually with equations such as 5 + 2 = 7 and 7 – 2 = 5. (Kindergartenstudents should see addition and subtraction equations, and student writing of equations in kindergarten is encouraged, but it is not required.) Studentschoose, 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 aftersome are taken away.Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. Theterms students should learn to use with increasing precision with this cluster are: join, separate, add and subtract.Common Core Standard Unpacking What do these standards mean a child will know and be able to do?K.OA.1 Represent addition and K.OA.1 asks students to demonstrate the understanding of how objects can be joined (addition) and separatedsubtraction with objects, fingers, mental (subtraction) by representing addition and subtraction situations in various ways. This objective is primarilyimages, drawings2, sounds (e.g., claps), focused on understanding the concept of addition and subtraction, rather than merely reading and solving additionacting out situations, verbal and subtraction number sentences (equations).explanations, expressions, or equations.2 Drawings need not show details, butshould show the mathematics in theproblem. (This applies whereverdrawings are mentioned in theStandards.)K.OA.2 Solve addition and subtraction K.OA.2 asks students to solve problems presented in a story format (context) with a specific emphasis on usingword problems, and add and subtract objects or drawings to determine the solution. This objective builds upon their understanding of addition andwithin 10, e.g., by using objects or subtraction from K.OA.1, to solve problems. Once again, numbers should not exceed 10.drawings to represent the problem. Teachers should be cognizant of the three types of problems. There are three types of addition and subtraction problems: Result Unknown, Change Unknown, and Start Unknown. These types of problems become increasingly difficult for students. Research has found that Result Unknown problems are easier than Change and Start Unknown problems. Kindergarten students should have experiences with all three types of problems. The level of difficulty can be decreased by using smaller numbers (up to 5) or increased by using larger numbers (up to 10). Please see Appendix, Table 1 for additional examples. Kindergarten Mathematics ● Unpacked Content page 9
- 10. Addition Examples: Result Unknown: There are 3 Change Unknown: There are 3 Start Unknown: There are some students on the playground. students on the playground. Some students on the playground. Four Four more students showed up. more students show up. There are more students came. There are now 7 How many students are there now 7 students. How many students students. How many students were on now? came? the playground at the beginning? (3+4 = __) (3+ __ = 7) (__ + 4 = 7)K.OA.3 Decompose numbers less than K.OA.3 asks students to understand that a set of (5) object can be broken into two sets (3 and 2) and still be theor equal to 10 into pairs in more than same total amount (5). In addition, this objective asks students to realize that a set of objects (5) can be broken inone way, e.g., by using objects or multiple ways (3 and 2; 4 and 1). Thus, when breaking apart a set (decomposing), students develop thedrawings, and record each understanding that a smaller set of objects exists within that larger set (inclusion). This should be developed indecomposition by a drawing or equation context before moving into how to represent decomposition with symbols (+, -, =).(e.g., 5 = 2 + 3 and 5 = 4 + 1). Example: “Bobby Bear is missing 5 buttons on his jacket. How many ways can you use blue and red buttons to finish his jacket? Draw a picture of all your ideas. Students could draw pictures of: 4 blue and 1 red button 3 blue and 2 red buttons 2 blue and 3 red buttons 1 blue and 4 red buttons After the students have had numerous experiences with decomposing sets of objects and recording with pictures and numbers, the teacher eventually makes connections between the drawings and symbols:5=4+1, 5=3+2, 5=2+3, and 5=1+4 The number sentence only comes after pictures or work with manipulatives, and students should never give the number sentence without a mathematical representation.K.OA.4 For any number from 1 to 9, KOA.4 builds upon the understanding that a number can be decomposed into parts (K.OA.3). Once studentsfind the number that makes 10 when have had experiences breaking apart ten into various combinations, this asks students to find a missing part of 10.added to the given number, e.g., byusing objects or drawings, and recordthe answer with a drawing or equation. Kindergarten Mathematics ● Unpacked Content page 10
- 11. Example below: “A full case of juice boxes has 10 boxes. There are only 6 boxes in this case. How many juice boxes are missing? Student 1: Student 2: Student 3: Using a Ten-Frame Think Addition Basic Fact I used 6 counters for the 6 boxes of juice “I counted out 10 cubes because I I know that it‟s 4 still in the case. There are 4 blank spaces, knew there needed to be ten. I because 6 and 4 is so 4 boxes have been removed. This makes pushed these 6 over here because the same amount as sense since 6 and 4 more equals 10. they were in the container. These 10. are left over. So there‟s 4 missing.”K.OA.5 Fluently add and subtract K.OA.5 uses the word fluently, which means accuracy (correct answer), efficiency (a reasonable amount ofwithin 5. steps), and flexibility (using strategies such as the distributive property). Fluency is developed by working with many different kinds of objects over an extended amount of time. This objective does not require students to instantly know the answer. Traditional flash cards or timed tests have not been proven as effective instructional strategies for developing fluency.Number and Operations in Base Ten K.NBTCommon Core Standard and ClusterWork with numbers 11–19 to gain foundations for place value.Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. Theterms students should learn to use with increasing precision with this cluster are: tens, ones Common Core Standard Unpacking What do these standards mean a child will know and be able to do?K.NBT.1 Compose and decompose K.NBT.1 is the first time that students move beyond the number 10 with representations, such as objectsnumbers from 11 to 19 into ten ones (manipulatives) or drawings. The spirit of this standard is that students separate out a set of 11-19 objects into aand some further ones, e.g., by using group of ten objects with leftovers. This ability is a pre-cursor to later grades when they need to understand theobjects or drawings, and record each complex concept that a group of 10 objects is also one ten (unitizing). Ample experiences with ten frames will Kindergarten Mathematics ● Unpacked Content page 11
- 12. composition or decomposition by a help solidify this concept. Research states that students are not ready to unitize until the end of first grade.drawing or equation (e.g., 18 = 10 + 8); Therefore, this work in Kindergarten lays the foundation of composing tens and recognizing leftovers.understand that these numbers arecomposed of ten ones and one, two, Example below:three, four, five, six, seven, eight, or Teacher: “Please count out 15 chips.”nine ones. Student: Student counts 15 counters (chips or cubes). Teacher: “Do you think there is enough to make a group of ten chips? Do you think there might be some chips leftover?” Student: Student answers. Teacher: “Use your counters to find out.” Student: Student can either fill a ten frame or make a stick of ten connecting cubes. They answer, “There is enough to make a group of ten.” Teacher: How many leftovers do you have? Student: Students say, “I have 5 left over.” Teacher: How could we use words and/or numbers to show this? Student: Students might say “Ten and five is the same amount as 15”, “15 = 10 + 5”Measurement and Data K.MDCommon Core Standard and ClusterDescribe and compare measurable attributes.Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. Theterms students should learn to use with increasing precision with this cluster are: length, weight, heavy, long, more of, less of, longer, taller, shorter. Common Core Standard Unpacking What do these standards mean a child will know and be able to do?K.MD.1 Describe measurable attributes K.MD.1 calls for students to describe measurable attributes of objects, such as length, weight, size. For example,of objects, such as length or weight. a student may describe a shoe as “This shoe is heavy! It‟s also really long.” This standard focuses on usingDescribe several measurable attributes descriptive words and does not mean that students should sort objects based on attributes. Sorting appears later inof a single object. the Kindergarten standards. Kindergarten Mathematics ● Unpacked Content page 12
- 13. K.MD.2 Directly compare two objects K.MD.2 asks for direct comparisons of objects. Direct comparisons are made when objects are put next to eachwith a measurable attribute in common, other, such as two children, two books, two pencils. For example, a student may line up two blocks and say,to see which object has “more of”/“less “This block is a lot longer than this one.” Students are not comparing objects that cannot be moved and lined upof” the attribute, and describe the next to each other.difference.For example, directly compare the Through ample experiences with comparing different objects, children should recognize that objects should beheights of two children and describe matched up at the end of objects to get accurate measurements. Since this understanding requires conservation ofone child as taller/shorter. length, a developmental milestone for young children, children need multiple experiences to move beyond the idea that …. “Sometimes this block is longer than this one and sometimes it‟s shorter (depending on how I lay them side by side) and that‟s okay.” “This block is always longer than this block (with each end lined up appropriately).” Before conservation of length: The striped block is longer than the plain block when they are lined up like this. But when I move the blocks around, sometimes the plain block is longer than the striped block. After conservation of length: I have to line up the blocks to measure them. The plain block is always longer than the striped block.Common Core ClusterClassify objects and count the number of objects in each category. Common Core Standard Unpacking What do these standards mean a child will know and be able to do?K.MD.3 Classify objects into given K.MD.3 asks students to identify similarities and differences between objects (e.g., size, color, shape) and use thecategories; count the numbers of objects identified attributes to sort a collection of objects. Once the objects are sorted, the student counts the amount inin each category and sort the categories each set. Once each set is counted, then the student is asked to sort (or group) each of the sets by the amount inby count. each set. Kindergarten Mathematics ● Unpacked Content page 13
- 14. (Limit category counts to be less than or For example, when given a collection of buttons, the student separates the buttons into different piles based onequal to 10) color (all the blue buttons are in one pile, all the orange buttons are in a different pile, etc.). Then the student counts the number of buttons in each pile: blue (5), green (4), orange (3), purple (4). Finally, the student organizes the groups by the quantity in each group (Orange buttons (3), Green buttons next (4), Purple buttons with the green buttons because purple also had (4), Blue buttons last (5). This objective helps to build a foundation for data collection in future grades. In later grade, students will transfer these skills to creating and analyzing various graphical representations.Geometry K.GCommon Core Standard and ClusterIdentify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).This entire cluster asks students to understand that certain attributes define what a shape is called (number of sides, number of angles, etc.) and otherattributes do not (color, size, orientation). Then, using geometric attributes, the student identifies and describes particular shapes listed above. Throughoutthe year, Kindergarten students move from informal language to describe what shapes look like (e.g., “That looks like an ice cream cone!”) to more formalmathematical language (e.g., “That is a triangle. All of its sides are the same length”). In Kindergarten, students need ample experiences exploring variousforms of the shapes (e.g., size: big and small; types: triangles, equilateral, isosceles, scalene; orientation: rotated slightly to the left, „upside down‟) usinggeometric vocabulary to describe the different shapes. In addition, students need numerous experiences comparing one shape to another, rather thanfocusing on one shape at a time. This type of experience solidifies the understanding of the various attributes and how those attributes are different- orsimilar- from one shape to another.Students in Kindergarten typically recognize figures by appearance alone, often by comparing them to a known example of a shape, such as the triangle onthe left. For example, students in Kindergarten typically recognize that the figure on the left as a triangle, but claim that the figure on the right is not atriangle, since it does not have a flat bottom. The properties of a figure are not recognized or known. Students make decisions on identifying and describingshapes based on perception, not reasoning.Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. Theterms students should learn to use with increasing precision with this cluster are: squares, circles, triangles, rectangles, hexagons, cubes, cones,cylinders, spheres, above, below, beside, in front of, behind, next to, flat, solid, side, corner and equal. Kindergarten Mathematics ● Unpacked Content page 14
- 15. Common Core Standards Unpacking What do these standards mean a child will know and be able to do?K.G.1 Describe objects in the K.G.1 expects students to use positional words (such as those italicized above) to describe objects in theenvironment using names of shapes, environment. Kindergarten students need to focus first on location and position of two-and-three-dimensionaland describe the relative positions of objects in their classroom prior to describing location and position of two-and-three-dimension representations onthese objects using terms such as above, paper.below, beside, in front of, behind, andnext to.K.G.2 Correctly name shapes K.G.2 addresses students‟ identification of shapes based on known examples. Students at this level do not yetregardless of their orientations or recognize triangles that are turned upside down as triangles, since they don‟t “look like” triangles. Students needoverall size. ample experiences looking at and manipulating shapes with various typical and atypical orientations. Through these experiences, students will begin to move beyond what a shape “looks like” to identifying particular geometric attributes that define a shape.K.G.3 Identify shapes as two- K.G.3 asks students to identify flat objects (2 dimensional) and solid objects (3 dimensional). This standard candimensional (lying in a plane, “flat”) or be done by having students sort flat and solid objects, or by having students describe the appearance or thicknessthree dimensional (“solid”). of shapes.Common Core ClusterAnalyze, compare, create, and compose shapes. Common Core Standard Unpacking What do these standards mean a child will know and be able to do?K.G.4 Analyze and compare two- and K.G.4 asks students to note similarities and differences between and among 2-D and 3-D shapes using informalthree-dimensional shapes, in different language. These experiences help young students begin to understand how 3-dimensional shapes are composedsizes and orientations, using informal of 2-dimensional shapes (e.g.., The base and the top of a cylinder is a circle; a circle is formed when tracing alanguage to describe their similarities, sphere).differences, parts (e.g., number of sidesand vertices/“corners”) and otherattributes (e.g., having sides of equallength).K.G.5 Model shapes in the world by K.G.5 asks students to apply their understanding of geometric attributes of shapes in order to create given shapes.building shapes from components (e.g., For example, a student may roll a clump of play-doh into a sphere or use their finger to draw a triangle in the sandsticks and clay balls) and drawing table, recalling various attributes in order to create that particular shape.shapes.K.G.6 Compose simple shapes to form K.G.6 moves beyond identifying and classifying simple shapes to manipulating two or more shapes to create a Kindergarten Mathematics ● Unpacked Content page 15
- 16. larger shapes. For example, “Can you new shape. This concept begins to develop as students‟ first move, rotate, flip, and arrange puzzle pieces. Next,join these two triangles with full sides students use their experiences with puzzles to move given shapes to make a design (e.g., “Use the 7 tangramtouching to make a rectangle?” pieces to make a fox.”). Finally, using these previous foundational experiences, students manipulate simple shapes to make a new shape. Kindergarten Mathematics ● Unpacked Content page 16
- 17. Table 1 Common addition and subtraction situations1 Result Unknown Change Unknown Start Unknown Two bunnies sat Two bunnies were Some bunnies were on the grass. sitting on the grass. sitting on the grass. Three more Some more Three more bunnies bunnies hopped bunnies hopped hopped there. Then there. How many there. Then there there were five bunnies. Add to bunnies are on were five bunnies. How many bunnies the grass now? How many bunnies were on the grass 2+3=? hopped over to the before? first two? ?+3=5 2+?=5 Five apples were Five apples were Some apples were on on the table. I ate on the table. I ate the table. I ate two two apples. How some apples. Then apples. Then there were Take many apples are there were three three apples. How many from on the table now? apples. How many apples were on the table 5–2=? apples did I eat? before? ? – 2 = 3 5–?=3 Total Unknown Addend Unknown Both Addends Unknown2 Three red apples Five apples are on Grandma has five and two green the table. Three are flowers. How many can Put apples are on the red and the rest are she put in her red vaseTogether/ table. How many green. How many and how many in her Take apples are on the apples are green? blue vase? Apart 3 table? 3 + ? = 5, 5 – 3 = ? 5 = 0 + 5, 5 = 5 + 0 3+2=? 5 = 1 + 4, 5 = 4 + 1 5 = 2 + 3, 5 = 3 + 2Kindergarten Mathematics ● Unpacked Content
- 18. Difference Bigger Unknown Smaller Unknown Unknown (“How many (Version with (Version with “more”): more?” version): “more”): Julie has three more Lucy has two Julie has three apples than Lucy. Julie apples. Julie has more apples than has five apples. How five apples. How Lucy. Lucy has many apples does Lucy many more two apples. How have? apples does Julie many apples does have than Lucy? Julie have? (Version with “fewer”): Lucy has 3 fewer applesCompare4 (“How many (Version with than Julie. Julie has five fewer?” version): “fewer”): apples. How many Lucy has two Lucy has 3 fewer apples does Lucy have? apples. Julie has apples than Julie. 5 – 3 = ?, ? + 3 = 5 five apples. How Lucy has two many fewer apples. How many apples does Lucy apples does Julie have than Julie? have? 2 + ? = 5, 5 – 2 = 2 + 3 = ?, 3 + 2 = ? ?2 These take apart situations can be used to show all the decompositions of a given number. The associated equations,which have the total on the left of the equal sign, help children understand that the = sign does not always mean makesor results in but always does mean is the same number as.3 Either addend can be unknown, so there are three variations of these problem situations. Both Addends Unknown is aproductive extension of this basic situation, especially for small numbers less than or equal to 10.4 For the Bigger Unknown or Smaller Unknown situations, one version directs the correct operation (the version usingmore for the bigger unknown and using less for the smaller unknown). The other versions are more difficult.1 Adapted from Box 2-4 of Mathematics Learning in Early Childhood, National Research Council (2009, pp. 32, 33).Kindergarten Mathematics ● Unpacked Contentpage 18

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