Kindergarten MathematicsRelationship Between NumbersUnit 2 Curriculum Map: November 14th, 2016- January 26th, 2016

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ORANGE PUBLIC SCHOOLSOFFICE OF CURRICULUM AND INSTRUCTIONOFFICE OF MATHEMATICS

Kindergarten Unit II Table of ContentsI. Mathematics Mission Statement p. 2II. Mathematical Teaching Practices p. 3III. Mathematical Goal Setting p. 4IV. Reasoning and Problem Solving p. 6V. Mathematical Representations p. 7VI. Mathematical Discourse p. 9VII. Conceptual Understanding p. 14VIII. Evidence of Student Thinking p. 15IX. Kindergarten Unit II NJSLS p. 17X. Eight Mathematical Practices p. 24XI. Ideal Math Block p. 26XII. Math Workstations p. 27XIII. Math In Focus Lesson Structure p. 30XIX. Ideal Math Block Planning Template p. 33XX. Planning Calendar p. 36XXI. Instructional and Assessment Framework p. 38XXII. PLD Rubric p. 43XXIII. Data Driven Instruction p. 44XXIV. Math Portfolio Expectations p. 461

XXV. Resources p. 48Office of Mathematics Mission StatementThe Office of Mathematics exists to provide the students it serves with a mathematical ‘lens’-- allowing them to better access the world with improved decisiveness, precision, and dexterity; facilities attained as students develop a broad and deep understanding of mathematical content. Achieving this goal defines our work - ensuring that students are exposed to excellence via a rigorous, standards-driven mathematics curriculum, knowledgeable and effective teachers, and policies that enhance and support learning. Office of Mathematics Objective By the year 2021, Orange Public School students will demonstrate improved academic achievement as measured by a 25% increase in the number of students scoring at or above the district’s standard for proficient (college ready (9-12); on track for college and career (K-8)) in Mathematics.Rigorous, Standards-Driven Mathematics CurriculumThe Grades K-8 mathematics curriculum was redesigned to strengthen students’ procedural skills and fluency while developing the foundational skills of mathematical reasoning and problem solving that are crucial to success in high school mathematics. Our curriculum maps are Unit Plans that are in alignment with the New Jersey Student Learning Standards for Mathematics. Office of Mathematics Department HandbookResearch tells us that teacher knowledge is one of the biggest influences on classroom atmosphere and student achievement (Fennema & Franke, 1992). This is because of the daily tasks of teachers, interpreting someone else’s work, representing and forging links between ideas in multiple forms, developing alternative explanations, and choosing usable definitions. (Ball, 2003; Ball, et al., 2005; Hill & Ball, 2009). As such, the Office of Mathematics Department Handbook and Unit Plans were intentionally developed to facilitate the daily work of our teachers; providing the tools necessary for the alignment between curriculum, instruction, and assessment. These document helps to (1) communicate the shifts (explicit and implicit) in the New Jersey Student Learning Standards for elementary and secondary mathematics (2) set course expectations for each of our courses of study and (3) encourage

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teaching practices that promote student achievement. These resources are accessible through the Office of Mathematics website.Curriculum Unit PlansDesigned to be utilized as a reference when making instructional and pedagogical decisions, Curriculum Unit Plans include but are not limited to standards to be addressed each unit, recommended instructional pacing, best practices, as well as an assessment framework.

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Mathematical Teaching Practices

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Mathematical Goal Setting: What are the math expectations for student learning? In what ways do these math goals focus the teacher’s interactions with students throughout the lesson?

Learning Goals should: Clearly state what students are to learn and understand about mathematics as the result of instruction. Be situated within learning progressions. Frame the decisions that teachers make during a lesson.

Example:New Jersey Student Learning Standards:K.OA.1Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.K.OA.2Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.K.OA.5Fluently add and subtract within 5.Learning Goal(s):Students will use multiple representations to solve addition and/or subtraction situations (K.OA.1-2) and explain their solution paths.Student Friendly Version:We are learning to act out and solve addition and/ or subtraction situations.We are will also be able to explain how we solved different situations.

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Lesson Implementation:As students reason through their selected solution paths, educators use of questioning facilitates the accomplishment of the identified math goal. Students’ level of understanding becomes evident in what they produce and are able to communicate. Students can also assess their level of goal attainment and that of their peers through the use of a student friendly rubric (MP3).

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Student Name: __________________________________________ Task: ______________________________ School: ___________ Teacher: ______________ Date: ___________

“I CAN…..”STUDENT FRIENDLY RUBRIC SCORE …a start1 …getting there2 …that’s it3 WOW!4

Understand I need help. I need some help. I do not need help. I can help a classmate.  

Solve   I am unable to use a strategy. I can start to use a strategy. I can solve it more than one way.I can use more than one strategy and talk about how they get to the same answer.

 

Say or Write I am unable to say or write.I can write or say some of what I did.

I can write and talk about what I did.I can write or talk about why I did it.

I can write and say what I did and why I did it. 

 Draw I am not able to draw or show my I can draw, but not show my I can draw and show my thinking I can draw, show and talk about  7

or Show  thinking. thinking;orI can show but not draw my thinking;my thinking.

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Reasoning and Problem Solving Mathematical TasksThe benefits of using formative performance tasks in the classroom instead of multiple choice, fill in the blank, or short answer questions have to do with their abilities to capture authentic samples of students' work that make thinking and reasoning visible. Educators’ ability to differentiate between low-level and high-level demand task is essential to ensure that evidence of student thinking is aligned and targeted to learning goals. The Mathematical Task Analysis Guide serves as a tool to assist educators in selecting and implementing tasks that promote reasoning and problem solving.

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Use and Connection of Mathematical Representations

The Lesh Translation ModelEach oval in the model corresponds to one way to represent a mathematical idea. Visual: When children draw pictures, the teacher can learn more about what they understand about a particular mathematical idea and can use the different pictures that children create to provoke a discussion about mathematical ideas. Constructing their own pictures can be a powerful learning experience for children because they must consider several aspects of mathematical ideas that are often assumed when pictures are pre-drawn for students. Physical: The manipulatives representation refers to the unifix cubes, base-ten blocks, fraction circles, and the like, that a child might use to solve a problem. Because children can physically manipulate these objects, when used appropriately, they provide opportunities to compare relative sizes of objects, to identify patterns, as well as to put together representations of numbers in multiple ways.Verbal: Traditionally, teachers often used the spoken language of mathematics but rarely gave students opportunities to grapple with it. Yet, when students do have opportunities to express

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(Pictures)

(Written)

(Real Life Situations)

(Manipulatives)

(Communication)

their mathematical reasoning aloud, they may be able to make explicit some knowledge that was previously implicit for them. Symbolic: Written symbols refer to both the mathematical symbols and the written words that are associated with them. For students, written symbols tend to be more abstract than the other representations. I tend to introduce symbols after students have had opportunities to make connections among the other representations, so that the students have multiple ways to connect the symbols to mathematical ideas, thus increasing the likelihood that the symbols will be comprehensible to students. Contextual: A relevant situation can be any context that involves appropriate mathematical ideas and holds interest for children; it is often, but not necessarily, connected to a real-life situation. The Lesh Translation Model: Importance of ConnectionsAs important as the ovals are in this model, another feature of the model is even more important than the representations themselves: The arrows! The arrows are important because they represent the connections students make between the representations. When students make these connections, they may be better able to access information about a mathematical idea, because they have multiple ways to represent it and, thus, many points of access. Individuals enhance or modify their knowledge by building on what they already know, so the greater the number of representations with which students have opportunities to engage, the more likely the teacher is to tap into a student’s prior knowledge. This “tapping in” can then be used to connect students’ experiences to those representations that are more abstract in nature (such as written symbols). Not all students have the same set of prior experiences and knowledge. Teachers can introduce multiple representations in a meaningful way so that students’ opportunities to grapple with mathematical ideas are greater than if their teachers used only one or two representations.Concrete Pictorial Abstract (CPA) Instructional ApproachThe CPA approach suggests that there are three steps necessary for pupils to develop understanding of a mathematical concept. Concrete: “Doing Stage”: Physical manipulation of objects to solve math problems.

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Pictorial: “Seeing Stage”: Use of imaged to represent objects when solving math problems.Abstract: “Symbolic Stage”: Use of only numbers and symbols to solve math problems.CPA is a gradual systematic approach. Each stage builds on to the previous stage. Reinforcement of concepts are achieved by going back and forth between these representations

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Mathematical Discourse and Strategic QuestioningDiscourse involves asking strategic questions that elicit from students both how a problem was solved and why a particular method was chosen. Students learn to critique their own and others' ideas and seek out efficient mathematical solutions.While classroom discussions are nothing new, the theory behind classroom discourse stems from constructivist views of learning where knowledge is created internally through interaction with the environment. It also fits in with socio-cultural views on learning where students working together are able to reach new understandings that could not be achieved if they were working alone.Underlying the use of discourse in the mathematics classroom is the idea that mathematics is primarily about reasoning not memorization. Mathematics is not about remembering and applying a set of procedures but about developing understanding and explaining the processes used to arrive at solutions.Asking better questions can open new doors for students, promoting mathematical thinking and classroom discourse. Can the questions you're asking in the mathematics classroom be answered with a simple “yes” or “no,” or do they invite students to deepen their understanding?

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To help you encourage deeper discussions, here are 100 questions to incorporate into your instruction by Dr. Gladis Kersaint, mathematics expert and advisor for Ready Mathematics.

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Conceptual UnderstandingStudents demonstrate conceptual understanding in mathematics when they provide evidence that they can: recognize, label, and generate examples of concepts; use and interrelate models, diagrams, manipulatives, and varied representations of concepts; identify and apply principles; know and apply facts and definitions; compare, contrast, and integrate related concepts and principles; and recognize, interpret, and apply the signs, symbols, and terms used to represent concepts.

Conceptual understanding reflects a student's ability to reason in settings involving the careful application of concept definitions, relations, or representations of either.Procedural FluencyProcedural fluency is the ability to: apply procedures accurately, efficiently, and flexibly;

to transfer procedures to different problems and contexts; to build or modify procedures from other procedures; and to recognize when one strategy or procedure is more appropriate to apply than another.

Procedural fluency is more than memorizing facts or procedures, and it is more than understanding and being able to use one procedure for a given situation. Procedural fluency builds on a foundation of conceptual understanding, strategic reasoning, and problem solving (NGA Center & CCSSO, 2010; NCTM, 2000, 2014). Research suggests that once students have memorized and practiced procedures that they do not understand, they have less motivation to understand their meaning or the reasoning behind them (Hiebert, 1999). Therefore, the development of students’ conceptual understanding of procedures should precede and coincide with instruction on procedures. Math Fact Fluency: AutomaticityStudents who possess math fact fluency can recall math facts with automaticity. Automaticity is the ability to do things without occupying the mind with the low-level details required, allowing it to become an automatic response pattern or habit. It is usually the result of learning, repetition, and practice.K-2 Math Fact Fluency Expectation

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K.OA.5 Add and Subtract within 5.1.OA.6 Add and Subtract within 10.2.OA.2 Add and Subtract within 20.Math Fact Fluency: Fluent Use of Mathematical StrategiesFirst and second grade students are expected to solve addition and subtraction facts using a variety of strategies fluently.1.OA.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; and creating equivalent but easier or known sums.

2.NBT.7 Add and subtract within 1000, using concrete models or drawings and strategies based on:o place value, o properties of operations, and/oro the relationship between addition and subtraction;

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Evidence of Student ThinkingEffective classroom instruction and more importantly, improving student performance, can be accomplished when educators know how to elicit evidence of students’ understanding on a daily basis. Informal and formal methods of collecting evidence of student understanding enable educators to make positive instructional changes. An educators’ ability to understand the processes that students use helps them to adapt instruction allowing for student exposure to a multitude of instructional approaches, resulting in higher achievement. By highlighting student thinking and misconceptions, and eliciting information from more students, all teachers can collect more representative evidence and can therefore better plan instruction based on the current understanding of the entire class. Mathematical ProficiencyTo be mathematically proficient, a student must have: • Conceptual understanding: comprehension of mathematical concepts, operations, and relations; • Procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently, and appropriately; • Strategic competence: ability to formulate, represent, and solve mathematical problems; • Adaptive reasoning: capacity for logical thought, reflection, explanation, and justification; • Productive disposition: habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy.Evidence should:

Provide a window in student thinking; Help teachers to determine the extent to which students are reaching the math learning goals; and Be used to make instructional decisions during the lesson and to prepare for subsequent lessons.

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K-2 CONCEPT MAP

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Kindergarten Unit II In this Unit Students will:Counting and Cardinality

Count to 100 by ones and by tens Count forward beginning from a given number within the known sequence. Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20. Understand the relationship between numbers and quantities; Count to answer "how many Given a number from 1-20, count out that many objects. Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group. Compare two numbers between 1 and 10 presented as written numerals. Decompose numbers less than or equal to 10 into pairs in more than one way.

Operation and Algebraic Thinking Fluently add and subtract within 5.

Numbers and Operations in Base Ten 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 Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.

Geometry 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. Correctly name shapes regardless of their orientations or overall size. Identify shapes as two-dimensional (lying in a plane, "flat") or three-dimensional ("solid"). Analyze and compare two- and three-dimensional shapes, in different sizes and orientations. Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes. Compose simple shapes to form larger shapes.

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New Jersey Student Learning Standards: Counting and CardinalityK.CC.1 Count to 100 by ones and by tens

Students’ rote count by starting at one and counting to 100. When students count by tens they are 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 a given number within the known sequence (instead of having to begin at 1).

Students begin a rote forward counting sequence from a number other than 1. Thus, given the number 4, the student would count, “4, 5, 6, 7 …” This objective does not require recognition of numerals. It is focused on the rote number sequence 0-100.K.CC.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).

Students write the numerals 0-20 and use the written numerals 0-20 to represent the amount within a set. For example, if the student has counted 9 objects, then the written numeral “9” is recorded. 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. For example, if a student picks up the number card “13”, the student then creates a pile of 13 counters. While children may experiment with writing numbers beyond 20, this standard places emphasis on numbers 0-20. Due to varied development of fine motor and visual development, reversal of numerals is anticipated. While reversals should be pointed out to students and correct formation modeled in instruction, the emphasis of this standard is on the use of numerals to represent quantities rather than the correct handwriting formation of the actual numeral itself.

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K.CC.4Understand the relationship between numbers and quantities; connect counting to cardinality.

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.

Students count a set of objects and see sets and numerals in relationship to one another. These connections are higher-level skills that require students to analyze, reason about, and explain relationships between numbers and sets of objects. The expectation is that students are comfortable with these skills with the numbers 1-20 by the end of Kindergarten.Students implement correct counting procedures by pointing to one object at a time (one-to-one correspondence), using one counting word for every object (synchrony/ one-to-one tagging), while keeping track of objects that have and have not been counted. This is the foundation of counting.Students answer the question “How many are there?” by counting objects in a set and understanding that the last number stated when counting a set (…8, 9, 10) represents the total amount of objects: “There are 10 bears in this pile.” (cardinality). Since an important goal for children is to count with meaning, it is important to have children answer the question, “How many do you have?” after they count. Often times, children who have not developed cardinality will count the amount again, not realizing that the 10 they stated means 10 objects in all.Young children believe what they see. Therefore, they may believe that a pile of cubes that they counted may be more if spread apart in a line. As children move towards the developmental milestone of conservation of number, they develop the understanding that the number of objects does not change when the objects are moved, rearranged, or hidden. Children need many different experiences with counting objects, as well as maturation, before they can reach this developmental milestone.

K.CC.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 1-20, count out that many objects.In order to answer “how many?” students need to keep track of objects when counting. Keeping track is a method of counting that is used to count each item once and only once when determining how many. After numerous experiences with counting objects, along with the developmental understanding that a group of objects counted multiple times will remain the same amount, students recognize the need for keeping track in order to accurately determine “how many”. Depending on the

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amount of objects to be counted, and the students’ confidence with counting a set of objects, students may move the objects as they count each, point to each object as counted, look without touching when counting, or use a combination of these strategies. It is important that children develop a strategy that makes sense to them based on the realization that keeping track is important in order to get an accurate count, as opposed to following a rule, such as “Line them all up before you count”, in order to get the right answer.As children learn to count accurately, they may count a set correctly one time, but not another. Other times they may be able to keep track up to a certain amount, but then lose track from then on. Some arrangements, such as a line or rectangular array, are easier for them to get the correct answer but may limit their flexibility with developing meaningful tracking strategies, so providing multiple arrangements help children learn how to keep track. Since scattered arrangements are the most challenging for students, this standard specifies that students only count up to 10 objects in a scattered arrangement and count up to 20 objects in a line, rectangular array, or circle.New Jersey Student Learning Standards: Operations and Algebraic Thinking

K.OA.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).Students develop an understanding of part-whole relationships as they recognize that a set of objects (5) can be broken into smaller sub-sets (3 and 2) and still remain the total amount (5). In addition, this objective asks students to realize that a set of objects (5) can be broken in multiple ways (3 and 2; 4 and 1). Thus, when breaking apart a set (decompose), students use the understanding that a smaller set of objects exists within that larger set (inclusion).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 buttonsIn Kindergarten, students need ample experiences breaking apart numbers and using the vocabulary “and” & “same amount as” before symbols (+, =) and equations (5= 3 + 2) are introduced. If equations are used, a mathematical representation (picture, objects) needs to be present as well.

K.OA.5 Fluently add and subtract within 5.26

Students are fluent when they display accuracy (correct answer), efficiency (a reasonable amount of steps in about 3-5 seconds* without resorting to counting), and flexibility (using strategies such as the distributive property). Students develop fluency by understanding and internalizing the relationships that exist between and among numbers. Oftentimes, when children think of each “fact” as an individual item that does not relate to any other “fact”, they are attempting to memorize separate bits of information that can be easily forgotten. Instead, in order to fluently add and subtract, children must first be able to see sub-parts within a number (inclusion, K.CC.4.c).

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New Jersey Student Learning Standards: Numbers and Operations in Base Ten

K.NBT.1Compose 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.

Students explore numbers 11-19 using representations, such as manipulatives or drawings. Keeping each count as a single unit, kindergarteners use 10 objects to represent “10” rather than creating a unit called a ten (unitizing) as indicated in the First Grade CCSS standard 1.NBT.1a: 10 can be thought of as a bundle of ten ones — called a “ten.”Example:Teacher: “I have some chips here. Do you think they will fit on our ten frame? Why? Why Not?”Students: Share thoughts with one another.Teacher: “Use your ten frame to investigate.”Students: “Look. There’s too many to fit on the ten frame. Only ten chips will fit on it.”Teacher: “So you have some leftovers?”Students: “Yes. I’ll put them over here next to the ten frame.”Teacher: “So, how many do you have in all?”Student A: “One, two, three, four, five… ten, eleven, twelve, thirteen, fourteen. I have fourteen. Ten fit on and four didn’t.”Student B: Pointing to the ten frame, “See them- that’s 10… 11, 12, 13, 14. There’s fourteen.”Teacher: Use your recording sheet (or number sentence cards) to show what you found out.Student Recording Sheets Example:

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New Jersey Student Learning Standards: Measurement and DataK.MD.3 Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.(Limit category counts to be less than or equal to 10)

Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.

New Jersey Student Learning Standards: Geometry

K.G.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.Students locate and identify shapes in their environment. For example, a student may look at the tile pattern arrangement on the hall floor and say, “Look! I see squares! They are next to the triangle.” At first students may use informal names e.g., “balls,” “boxes,” “cans”. Eventually students refine their informal language by learning mathematical concepts and vocabulary and identify, compare, and sort shapes based on geometric attributes.*Students also use positional words (such as those italicized in the standard) to describe objects in the environment, developing their spatial reasoning competencies. Kindergarten students need numerous experiences identifying the location and position of actual two-and-three-dimensional objects in their classroom/school prior to describing location and position of two-and-three-dimension representations on paper.

*Progressions for the CCSS in Mathematics: Geometry, The Common Core Standards Writing Team, June 2012

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K.G.2 Correctly name shapes regardless of their orientations or overall size.

Through numerous experiences exploring and discussing shapes, students begin to understand that certain attributes define what a shape is called (number of sides, number of angles, etc.) and that other attributes do not (color, size, orientation). As the teacher facilitates discussions about shapes (“Is it still a triangle if I turn it like this?”), children question what they “see” and begin to focus on the geometric attributes. Kindergarten students typically do not yet recognize triangles that are turned upside down as triangles, since they don’t “look like” triangles. Students need ample experiences manipulating shapes and looking at 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-dimensional (lying in a plane, "flat") or three-dimensional ("solid").

Students identify objects as flat (2 dimensional) or solid (3 dimensional). As the teacher embeds the vocabulary into students’ exploration of various shapes, students use the terms two-dimensional and three-dimensional as they discuss the properties of various shapes.

K.G.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).31

Students relate one shape to another as they note similarities and differences between and among 2-D and 3-D shapes using informal language. For example, when comparing a triangle and a square, they note that they both are closed figures, have straight sides, but the triangle has 3 sides while the square has 4. Or, when building in the Block Center, they notice that the faces on the cube are all square shapes. Kindergarteners also distinguish between the most typical examples of a shape from obvious non-examples. For example: When identifying the triangles from a collection of shapes, a student circles all of the triangle examples from the non-examples.

K.G.5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.Students apply their understanding of geometric attributes of shapes in order to create given shapes. For example, students may roll a clump of play-doh into a sphere or use their finger to draw a triangle in the sand table, recalling various attributes in order to create that particular shape.

K.G.6 Compose simple shapes to form larger shapes. For example, “Can you join these two triangles with full sides touching to make a rectangle?”This concept begins to develop as students move, rotate, flip, and arrange puzzle pieces to complete a puzzle. Kindergarteners use their experiences with puzzles to use simple shapes to create different shapes. For example, when using basic shapes to create a picture, a student flips and turns triangles to make a rectangular house. Students also combine shapes to build pictures. They first use trial and error and gradually consider components.

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Eight Mathematical PracticesThe Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. 

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Make sense of problems and persevere in solving them

In Kindergarten, students learn that doing math involves solving problems and discussing how they solved them. Students will begin to explain the meaning of a problem, and look for ways to solve it. Kindergarteners will learn how to use objects and pictures to help them understand and solve problems. They will begin to check their thinking when the teacher asks them how they got their answer, and if the answer makes sense. When working in small groups or with a partner they will listen to the strategies of the group and will try different approaches.

2 Reason abstractly and quantitatively

Mathematically proficient students in Kindergarten make sense of quantities and the relationships while solving tasks. This involves two processes- decontextualizing and contextualizing. In Kindergarten, students represent situations by 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.34

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Construct viable arguments and critique the reasoning of others

Mathematically proficient students in Kindergarten accurately use mathematical terms to construct arguments and engage in discussions about problem solving strategies. For example, while solving the task, “There are 8 books on the shelf. If you take 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 then be able to construct an accurate argument about why they subtracted 3 form 8 rather than adding 8 and 3. Further, Kindergarten students are expected to examine a variety of problem solving strategies and begin to recognize the reasonableness of them, as well as similarities and differences among them.

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Model with mathematics

Mathematically proficient students in Kindergarten 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. 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.

5 Use appropriate tools strategically

Mathematically proficient students in Kindergarten have access to and use tools appropriately. These tools may include counters, place value (base ten) blocks, hundreds number boards, number lines, and concrete geometric shapes (e.g., pattern blocks, 3-d solids). Students should also have experiences with educational technologies, such as calculators, virtual manipulatives, and mathematical games that support conceptual understanding. 35

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.”

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Attend to precision

Mathematically proficient students in Kindergarten are precise in their communication, calculations, and measurements. In all 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.

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Look for and make use of structure

Mathematically proficient students in Kindergarten carefully look for patterns and structures in the number system and other areas of mathematics. While solving addition problems, students begin to recognize the commutative property, in that 1+4 = 5, 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 express regularity in repeated reasoning

Mathematically proficient students in Kindergarten begin to look for regularity in problem structures when solving mathematical tasks. Likewise, students begin composing and decomposing numbers in different ways. For example, in the task “There are 8 crayons in the box. Some are red and some are blue. 36

How many of each could there be?”Kindergarten students 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. For each solution, students repeated engage in the process of finding two numbers that can be joined to equal 8.

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Note: Place emphasis on the flow of the lesson in order to ensure the development of students’ conceptual understanding. Outline each essential component within lesson plans. Math Workstations may be conducted in the beginning of the block in order to utilize additional support staff. Recommended: 5-10 technology devices for use within TECHNOLOGY and FLUENCY workstations.

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EXPLORE: Partner / Small GroupMath In Focus Hands-On, Guided Practice, Let’s Explore

APPLY: INDEPENDENT PRACTICEMath In Focus Let’s Practice, Workbook, Reteach, Extra Practice, Enrichment

Getting Ready: Whole GroupAnchor Task: Math In Focus Learn

CENTERS/STATIONS: Pairs / Small Group/ Individual

DIFFERENTIATED activities designed to RETEACH, REMEDIATE, ENRICH student’s understanding of concepts.

Small Group Instruction

ProblemSolving

LabTechnology Lab

Math Journal

LabFluency Lab

Kindergarten Ideal Math Block Essential Components FLUENCY: Partner/Small Group

CONCRETE, PICTORIAL, and ABSTRACT approaches to support ARITHMETIC FLUENCY and FLUENT USE OF STRATEGIES.

SUMMARY: Whole GroupLesson Closure: Student Reflection; Real Life Connections to Concept

DISCOVER: Whole GroupProvides HANDS-ON work to allow children to ACT OUT or ENGAGE ACTIVELY

with the new MATH IDEA

5 min.

15-20 min.

50-60 min.

INVESTIGATE: Whole GroupChildren are invited to sing, clap, rhyme, and discuss colorful, playful scenes

presented in the BIG BOOK while the teacher SYSTEMATICALLY EMPLOYS and ELICITS related MATH TALK.

MATH WORKSTATIONSMath workstations allow students to engage in authentic and meaningful hands-on learning. They often last for several weeks, giving students time to reinforce or extend their prior instruction. Before students have an opportunity to use the materials in a station, introduce them to the whole class, several times. Once they have an understanding of the concept, the materials are then added to the work stations. Station Organization and Management SampleTeacher A has 12 containers labeled 1 to 12. The numbers correspond to the numbers on the rotation chart. She pairs students who can work well together, who have similar skills, and who need more practice on the same concepts or skills.  Each day during math work stations, students use the center chart to see which box they will be using and who their partner will be. Everything they need for their station will be in their box. Each station is differentiated. If students need more practice and experience working on numbers 0 to 10, those will be the only numbers in their box. If they are ready to move on into the teens, then she will place higher number activities into the box for them to work with.

In the beginning there is a lot of prepping involved in gathering, creating, and organizing the work stations. However, once all of the initial work is complete, the stations are easy to manage. Many of her stations stay in rotation for three or four weeks to give students ample opportunity to master the skills and concepts. Read Math Work Stations by Debbie Diller.

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In her book, she leads you step-by-step through the process of implementing work stations.

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MATH WORKSTATION INFORMATION CARD

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Math Workstation: _____________________________________________________________________ Time: _________________NJSLS.: ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Objective(s): By the end of this task, I will be able to:

________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Task(s):___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Exit Ticket: __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

MATH WORKSTATION SCHEDULE Week of: _________________

DAY TechnologyLab

Problem Solving Lab FluencyLab

Math Journal

Small Group Instruction

Mon. Group ____ Group ____ Group ____ Group ____BASED

ON CURRENT OBSERVATIONAL

DATA

Tues. Group ____ Group ____ Group ____ Group ____

Wed. Group ____ Group ____ Group ____ Group ____

Thurs. Group ____ Group ____ Group ____ Group ____

Fri. Group ____ Group ____ Group ____ Group ____

INSTRUCTIONAL GROUPINGGROUP A GROUP B

1 12 23 34 45 56 6

GROUP C GROUP D1 12 23 34 45 56 6

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Math In Focus Lesson StructureLESSON STRUCTURE RESOURCES COMMENTS

Chapter OpenerAssessing Prior Knowledge

The Pre Test serves as a diagnostic test of readiness of the upcoming chapter

Teacher MaterialsQuick CheckPre-Test (Assessment Book)Recall Prior Knowledge

Student MaterialsStudent Book (Quick Check); Copy of the Pre Test; Recall prior Knowledge

Recall Prior Knowledge (RPK) can take place just before the pre-tests are given and can take 1-2 days to front load prerequisite understanding

Quick Check can be done in concert with the RPK and used to repair student misunderstandings and vocabulary prior to the pre-test ; Students write Quick Check answers on a separate sheet of paper

Quick Check and the Pre Test can be done in the same block (See Anecdotal Checklist; Transition Guide)

Recall Prior Knowledge – Quick Check – Pre Test

Direct Involvement/EngagementTeach/Learn

Students are directly involved in making sense, themselves, of the concepts – by interacting the tools, manipulatives, each other, and the questions

Teacher Edition5-minute warm upTeach; Anchor Task

TechnologyDigi

OtherFluency Practice

The Warm Up activates prior knowledge for each new lessonStudent Books are CLOSED; Big Book is used in Gr. KTeacher led; Whole groupStudents use concrete manipulatives to explore conceptsA few select parts of the task are explicitly shown, but the majority is addressed through the hands-on, constructivist approach and questioning Teacher facilitates; Students find the solution

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DIRECT ENGAGEMENT

PRE TEST

Guided Learning and PracticeGuided Learning

Teacher EditionLearn

TechnologyDigi

Student BookGuided Learning PagesHands-on Activity

Students-already in pairs /small, homogenous ability groups; Teacher circulates between groups; Teacher, anecdotally, captures student thinking

Small Group w/Teacher circulating among groupsRevisit Concrete and Model Drawing; ReteachTeacher spends majority of time with struggling learners; some time with on level, and less time with advanced groupsGames and Activities can be done at this time

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Independent Practice

A formal formative …assessment

Teacher EditionLet’s Practice

Student BookLet’s Practice

Differentiation OptionsAll: WorkbookExtra Support: ReteachOn Level: Extra PracticeAdvanced: Enrichment

Let’s Practice determines readiness for Workbook and small group work and is used as formative assessment; Students not ready for the Workbook will use Reteach. The Workbook is continued as Independent Practice.Manipulatives CAN be used as a communications tool as needed.Completely IndependentOn level/advance learners should finish all workbook pages.

Extending the Lesson Math JournalProblem of the LessonInteractivitiesGames

Lesson Wrap Up Problem of the LessonHomework (Workbook , Reteach, or Extra Practice)

Workbook or Extra Practice Homework is only assigned when students fully understand the concepts (as additional practice)Reteach Homework (issued to struggling learners) should be checked the next day

End of Chapter Wrap Up and Post Test

Teacher EditionChapter Review/TestPut on Your Thinking Cap

Student WorkbookPut on Your Thinking Cap

Assessment BookTest Prep

Use Chapter Review/Test as “review” for the End of Chapter Test Prep. Put on your Thinking Cap prepares students for novel questions on the Test Prep; Test Prep is graded/scored.The Chapter Review/Test can be completed Individually (e.g. for homework) then

reviewed in class As a ‘mock test’ done in class and

doesn’t count As a formal, in class review where

teacher walks students through the questions

Test Prep is completely independent; scored/gradedPut on Your Thinking Cap (green border) serve as a capstone problem and are done just before the Test Prep and should be treated as Direct Engagement. By February, students should be doing the Put on Your Thinking Cap problems on their own

GUIDED LEARNING

INDEPENDENT PRACTICE

ADDITIONAL PRACTICE

POST TEST

TRANSITION LESSON STRUCTURE (No more than 2 days) Driven by Pre-test results, Transition Guide Looks different from the typical daily lesson

Transition Lesson – Day 1Objective:CPA Strategy/Materials Ability Groupings/Pairs (by Name)

Task(s)/Text Resources Activity/Description

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IDEAL MATH BLOCK LESSON PLANNING TEMPLATECC

SS &

OBJ

:(s)

Fluency:K.OA.5Strategy:

Tool(s):

Mat

h In

Foc

us/E

nGag

eNY

Getting Ready

Investigate

Discover

Explore

Apply

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Diffe

rent

iatio

n: M

ath

Wor

ksta

tions

Small Group Instruction

Tech. Lab

Problem Solving Lab

CCSS:K.OA.1K.OA.2

Fluency LabK.OA.5

Strategy:

Tool(s):

Math Journal

MP3: Construct viable arguments and critique the reasoning of others

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Summary/ Lesson Closure

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Danielson Framework for Teaching: Domain 1: Planning PreparationLesson Planning Support Tool______________________________________________________________________________________________________Component 1A: Knowledge of Content and PedagogyContent

Fluency Practice and Anchor Problem clearly outlined in lesson plans provide reinforcement of prerequisite knowledge/skills needed;Essentials question(s) and lesson objective(s) support learning of New Jersey Student Learning Standards grade level expectations;PedagogyDaily fluency practice is clearly outlined in lesson plans;Multiple strategies are evident within lesson plans;Mathematical tools outlined within lesson plans;___________________________________________________________________________________________________________________________Component 1B: Knowledge of StudentsIntentional Student Grouping is evident within lesson plans: Independent Practice: Which students will work on:

MIF Re-TeachMIF PracticeMIF Extra PracticeMIF EnrichmentMath Workstations: Which students will work in:Fluency LabTechnology LabMath JournalProblem Solving Lab

Component 1C: Setting Instructional OutcomesLesson plan objectives are aligned to one or more New Jersey Student Standards for LearningConnections made to previous learningOutcomes: student artifacts are differentiated

Component 1D: Demonstrating Knowledge of ResourcesDistrict Approved Programs: Use Math In Focus/EnGageNY/Go Math resources are evidentTechnology: Technology used to help students understand the lesson objective is evident Students use technology to gain an understanding of the lesson objectiveSupplemental Resources: Integration of additional materials evident (Math Workstations)________________________________________________________________________________________________________________________Component 1E: Designing Coherent InstructionLesson Plans support CONCEPTUAL UNDERSTANDING

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Lesson Plans show evidences of CONCRETE, PICTORIAL, and ABSTRACT representationAlignment between OBJECTIVES, APPLICATION, and ASSESSMENT evident___________________________________________________________________________________________________________________________Component 1F: Assessing Student LearningLesson Plans include: Focus Question/Essential Understanding Anchor Problem Checks for Understanding Demonstration of Learning (Exit Ticket)

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Planning Calendar November 2016Monday Tuesday Wednesday Thursday Friday1 2 3 4

7 8 9 10 1114 15 16 17 1821 22 23 24 25

28 29 30December 2016

Monday Tuesday Wednesday Thursday Friday1 MIF CH.4 AssessmentMIF Ch. 4 Performance Task

2

5 6 7 8 9

12 13 14 15 16 MIF CH.6 AssessmentMIF Ch. 6 Performance Task19 20 21 22 23

26 27 28 29 30

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Planning Calendar January 2017

Monday Tuesday Wednesday Thursday Friday2 3 4 5 6

9 10 MIF CH.7 AssessmentMIF Ch. 7 Performance Task

11 12 13

16 17 18 19 2023 24 25 26 27

30 31February 2017

Monday Tuesday Wednesday Thursday Friday1 2 3

6 7 8 9 10

13 14 15 16 1720 21 22 23 24 27 28

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Math Workstation: Small Group Instruction: Administer K.CC and K.OA.5 Mid-Year AssessmentMath Workstation: Small Group Instruction: Administer K.CC and K.OA.5 Mid-Year Assessment

Kindergarten Unit 2 Instructional and Assessment FrameworkRecommended Activities CCSS Notes November 14th, 2016 Math In Focus Ch. 4 Lesson 1Composing and Decomposing 5 K.CC.1K.CC.2 K.CC.3 K.CC.4 K.CC.5 K.CC.6

K.OA.3K.OA.4

November 15, 2016 Math In Focus Ch. 4 Lesson 2Counting and Ordering Up to 10November 16, 2016 Math In Focus Ch. 4 Lesson 3Counting OnNovember 17, 2016 Math In Focus Ch. 4 Lesson 3Counting OnNovember 18, 2016 Math In Focus Ch. 4 Lesson 4Same Number and MoreNovember 21, 2016 Math In Focus Ch. 4 Lesson 4Same Number and MoreNovember 22, 2016 Math In Focus Ch. 4 Lesson 4Same Number and MoreNovember 23, 2016 Math In Focus Ch. 4 Lesson 5Fewer ThanNovember 28, 2016 Math In Focus Ch. 4 Lesson 5Fewer ThanNovember 29, 2016 Math In Focus Ch. 4 Lesson 6How Many in AllNovember 30, 2016 Math In Focus Ch. 4 Lesson 6How Many in AllDecember 1st , 2016 MIF Ch. 4 AssessmentMIF Ch. 4 Performance TaskDecember 2nd , 2016 Math In Focus Ch. 6 Lesson 1 All About 10 K.CC.1K.CC.2

K.CC.6K.CC.7December 5th , 2016 Math In Focus Ch. 6 Lesson 1 All About 10December 6th , 2016 Math In Focus Ch. 6 Lesson 2Number 10 to 12December 7th , 2016 Math In Focus Ch. 6 Lesson 2Number 10 to 12

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December 8th, 2016 Math In Focus Ch. 6 Lesson 3Number 13 to 16December 9th , 2016 Math In Focus Ch. 6 Lesson 3Number 13 to 16December 12th, 2016 Math In Focus Ch. 6 Lesson 4Number 17 to 20December 13th, 2016 Math In Focus Ch. 6 Lesson 4Number 17 to 20December 14th , 2016 Math In Focus Ch. 6 Lesson 5Compare and OrderDecember 15th, 2016 Math In Focus Ch. 6 Lesson 5Compare and OrderDecember 16th, 2016 MIF Ch. 6 AssessmentMIF Ch. 6 Performance TaskDecember 19th, 2016 Math In Focus Ch. 7 Lesson 1Solid Shapes K.G.1K.G.2K.G.3K.G.4K.G.5K.G.6December 20th, 2016 Math In Focus Ch. 7 Lesson 1Solid ShapesDecember 21th, 2016 Math In Focus Ch. 7 Lesson 2Flat Shapes in Solid ShapesDecember 22th, 2016 Math In Focus Ch. 7 Lesson 2Flat Shapes in Solid ShapesDecember 23th, 2016 Math In Focus Ch. 7 Lesson 3Flat ShapesJanuary 3rd, 2017 Math In Focus Ch. 7 Lesson 3Flat ShapesJanuary 4th, 2017 Math In Focus Ch. 7 Lesson 4Flat Shapes PicturesJanuary 5th, 2017 Math In Focus Ch. 7 Lesson 4Flat Shapes PicturesJanuary 6th, 2017 Math In Focus Ch. 7 Lesson 5Shape PatternsJanuary 9th, 2017 Math In Focus Ch. 7 Lesson 5Shape PatternsJanuary 10th, 2017 MIF Ch. 7 AssessmentMIF Ch. 7 Performance Task

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January 11th, 2017 Math In Focus Ch. 8 Lesson 1Counting by 2

K.CC.1K.CC.2 K.CC.3 K.CC.4 K.CC.5 K.CC.6

January 12th, 2017 Math In Focus Ch. 8 Lesson 1Counting by 2January 13th, 2017 Math In Focus Ch. 8 Lesson 2Counting by 5January 17th, 2017 Math In Focus Ch. 8 Lesson 2Counting by 5

Math Workstation: Small Group InstructionAdminister K.CC and K.OA.5 Mid-Year AssessmentJanuary 18th, 2017 Math In Focus Ch. 8 Lesson 3Counting by 10s to 100

Math Workstation: Small Group InstructionAdminister K.CC and K.OA.5 Mid-Year AssessmentJanuary 19th, 2017 Math In Focus Ch. 8 Lesson 3Counting by 10s to 100

Math Workstation: Small Group InstructionAdminister K.CC and K.OA.5 Mid-Year AssessmentJanuary 20th, 2017 Math In Focus Ch. 8 Lesson 4Numbers 20 to 49

Math Workstation: Small Group InstructionAdminister K.CC and K.OA.5 Mid-Year AssessmentJanuary 23rd , 2017 Math In Focus Ch. 8 Lesson 4Numbers 20 to 49

Math Workstation: Small Group InstructionAdminister K.CC and K.OA.5 Mid-Year AssessmentJanuary 24th, 2017 Math In Focus Ch. 8 Lesson 4Numbers 20 to 49

Math Workstation: Small Group InstructionAdminister K.CC and K.OA.5 Mid-Year AssessmentJanuary 25th, 2017 Math In Focus Ch. 8 Lesson 5Numbers 50 to 79

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Math Workstation: Small Group InstructionAdminister K.CC and K.OA.5 Mid-Year AssessmentJanuary 26th, 2017 Math In Focus Ch. 8 Lesson 5: Numbers 50 to 79 END OF MP 2

Math Workstation: Small Group InstructionAdminister K.CC and K.OA.5 Mid-Year AssessmentJanuary 27th, 2017 Math In Focus Ch. 8 Lesson 5: Numbers 50 to 79

Math Workstation: Small Group InstructionAdminister K.CC and K.OA.5 Mid-Year Assessment

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New Jersey Student Learning Standards Kindergarten Mathematics Student Fluency Progress Monitoring Tool NJSLS: K.OA.5: Add and subtract within 5.Student Name: _________________________________________ School: ___________________ Teacher: ___________________Student Conference Note:Student: Shade in each box upon mastery of fact.Teacher: Enter the date mastery was achieved for each fact in the upper left corner.

AUTOMATICITYAddition Facts within 5

5 4 3 2 1 00+55+0 0+44+0 0+33+0 0+22+0 0+11+0 0+01+44+1 1+33+1 1+22+1 1+12+33+2 2+2

Generalizations: # + 0 = #; (Additive Identity Property) # + 1 = next # in counting sequence; A group of numbers can be added in any order and their sum will always be the same. (Commutative Property of Addition)

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New Jersey Student Learning Standards Kindergarten Mathematics Student Fluency Progress Monitoring Tool NJSLS: K.OA.5: Add and subtract within 5.Student Name: _________________________________________ School: ___________________ Teacher: ___________________Student Conference Note:Student: Shade in each box upon mastery of fact.Teacher: Enter the date mastery was achieved for each fact.

AUTOMATICITYSubtraction Facts within 55 4 3 2 1 010-5 10-6 10-7 10-8 10-9 10-109-4 9-5 9-6 9-7 9-8 9-98-3 8-4 8-5 8-6 8-7 8-87-2 7-3 7-4 7-5 7-6 7-76-1 6-2 6-3 6-4 6-5 6-65-0 5-1 5-2 5-3 5-4 5-54-0 4-1 4-2 4-3 4-43-0 3-1 3-2 3-32-0 2-1 2-21-0 1-10-0Subtraction Generalizations:

# - 0 = #; # - # = 0; # - 1 = previous number in counting sequence;

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Kindergarten PLD RubricGot ItEvidence shows that the student essentially has the target concept or big math idea.Not There YetStudent shows evidence of a major misunderstanding, incorrect concepts or procedure, or a failure to engage in the task.PLD Level 5: 100%Distinguished command PLD Level 4: 89%Strong Command PLD Level 3: 79%Moderate Command PLD Level 2: 69%Partial Command PLD Level 1: 59%Little CommandStudent work shows distinguished levels of understanding of the mathematics.

Student constructs and communicates a complete response based on explanations/reasoning using the: Tools:

o Manipulativeso Five Frameo Ten Frameo Number Lineo Part-Part-Whole Model

Strategies:o Drawingso Counting Allo Count On/Backo Skip Countingo Making Teno Decomposing Number

Precise use of math vocabularyResponse includes an efficient and logical progression of mathematical reasoning

Student work shows strong levels of understanding of the mathematics. Student constructs and communicates a complete response based on explanations/reasoning using the: Tools:

o Manipulativeso Five Frameo Ten Frameo Number Lineo Part-Part-Whole Model

Strategies:o Drawingso Counting Allo Count On/Backo Skip Countingo Making Teno Decomposing Number

Precise use of math vocabularyResponse includes a logical progression of mathematical reasoning

Student work shows moderate levels of understanding of the mathematics. Student constructs and communicates a complete response based on explanations/reasoning using the: Tools:

o Manipulativeso Five Frameo Ten Frameo Number Lineo Part-Part-Whole Model

Strategies:o Drawingso Counting Allo Count On/Backo Skip Countingo Making Teno Decomposing Number

Precise use of math vocabularyResponse includes a logical but incomplete progression of

Student work shows partial understanding of the mathematics. Student constructs and communicates an incomplete response based on student’s attempts of explanations/ reasoning using the: Tools:

o Manipulativeso Five Frameo Ten Frameo Number Lineo Part-Part-Whole Model

Strategies:o Drawingso Counting Allo Count On/Backo Skip Countingo Making Teno Decomposing Number

Precise use of math vocabularyResponse includes an incomplete or illogical progression of

Student work shows little understanding of the mathematics. Student attempts to constructs and communicates a response using the: Tools:

o Manipulativeso Five Frameo Ten Frameo Number Lineo Part-Part-Whole Model

Strategies:o Drawingso Counting Allo Count On/Backo Skip Countingo Making Teno Decomposing Number

Precise use of math vocabularyResponse includes limited evidence of the progression of mathematical reasoning and understanding.

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and understanding. and understanding. mathematical reasoning and understanding. Contains minor errors.mathematical reasoning and understanding.

5 points 4 points 3 points 2 points 1 point

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DATA DRIVEN INSTRUCTIONFormative assessments inform instructional decisions. Taking inventories and assessments, observing reading and writing behaviors, studying work samples and listening to student talk are essential components of gathering data. When we take notes, ask questions in a student conference, lean in while a student is working or utilize a more formal assessment we are gathering data. Learning how to take the data and record it in a meaningful way is the beginning of the cycle.Analysis of the data is an important step in the process. What is this data telling us? We must look for patterns, as well as compare the notes we have taken with work samples and other assessments. We need to decide what are the strengths and needs of individuals, small groups of students and the entire class. Sometimes it helps to work with others at your grade level to analyze the data. Once we have analyzed our data and created our findings, it is time to make informed instructional decisions. These decisions are guided by the following questions: What mathematical practice(s) and strategies will I utilize to teach to these needs? What sort of grouping will allow for the best opportunity for the students to learn what it is I see as a need? Will I teach these strategies to the whole class, in a small guided group or in an individual conference? Which method and grouping will be the most effective and efficient? What specific objective(s) will I be teaching?

Answering these questions will help inform instructional decisions and will influence lesson planning.Then we create our instructional plan for the unit/month/week/day and specific lessons. It’s important now to reflect on what you have taught. Did you observe evidence of student learning through your checks for understanding, and through direct application in student work?

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What did you hear and see students doing in their reading and writing? Now it is time to begin the analysis again.

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Data Analysis Form School: __________________ Teacher: __________________________ Date: _______________Assessment: ____________________________________________ NJSLS: _____________________________________________________ GROUPS (STUDENT INITIALS) SUPPORT PLAN PROGRESSMASTERED (86% - 100%) (PLD 4/5):

DEVELOPING (67% - 85%) (PLD 3):

INSECURE (51%-65%) (PLD 2):

BEGINNING (0%-50%) (PLD 1):

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MATH PORTFOLIO EXPECTATIONSThe Student Assessment Portfolios for Mathematics are used as a means of documenting and evaluating students’ academic growth and development over time and in relation to the CCSS-M. Student Assessment Portfolios differ from student work folders in that they will contain tasks aligned specifically to the SGO focus. The September task entry(-ies) will reflect the prior year content and can serve as an additional baseline measure. All tasks contained within the Student Assessment Portfolios are “practice forward” (closely aligned to the Standards for Mathematical Practice). Four (4) or more additional tasks will be included in the Student Assessment Portfolios for Student Reflection and will be labeled as such. In March – June, the months extending beyond the SGO window, tasks will shift from the SGO focus to a focus on the In-depth Opportunities for each grade.K-2 GENERAL PORTFOLIO REQUIREMENTS As a part of last year’s end of year close-out process, we asked that student portfolios be ‘purged’; retaining a few artifacts and self-reflection documents that would transition with them to the next grade. In this current year, have students select 2-3 pieces of prior year’s work to file in the Student Assessment Portfolio. Tasks contained within the Student Assessment Portfolios are “practice forward” and denoted as “Individual”, “Partner/Group”, and “Individual w/Opportunity for Student Interviews1. Each Student Assessment Portfolio should contain a “Task Log” that documents all tasks, standards, and rubric scores aligned to the performance level descriptors (PLDs). Student work should be attached to a completed rubric; teacher feedback on student work is expected. Students will have multiple opportunities to revisit certain standards. Teachers will capture each additional opportunity “as a new and separate score” in the task log and in Genesis. A 2-pocket folder for each Student Assessment Portfolio is recommended.1 The Mathematics Department will provide guidance on task selection, thereby standardizing the process across the district and across grades/courses.

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All Student Assessment Portfolio entries should be scored and recorded in Genesis as an Authentic Assessment grade (25%)2. All Student Assessment Portfolios must be clearly labeled, maintained for all students, inclusive of constructive teacher and student feedback and accessible for administrator reviewMATHEMATICS PORTFOLIO: END OF YEAR REQUIREMENTSAt the start of the school year, you were provided with guidelines for helping students maintain their Mathematics Portfolios whereby students added artifacts that documented their growth and development over time. Included in the portfolio process was the opportunity for students to reflect on their thinking and evaluate what they feel constitutes “quality work.” As a part of the end of year closeout process, we are asking that you work with your students to help them ‘purge’ their current portfolios and retain the artifacts and self-reflection documents that will transition with them to the next grade.GRADES K-2Purging and Next-Grade TransitioningDuring the third (3rd) week of June, give students the opportunity to review and evaluate their portfolio to date; celebrating their progress and possibly setting goals for future growth. During this process, students will retain ALL of their current artifacts in their Mathematics Portfolios. The Student Profile Sheet from the end of year assessment should also be included in the student math portfolio. In the upcoming school year, after the new teacher has reviewed the portfolios, students will select 1-2 pieces to remain in the portfolio and take the rest home.

2 The Mathematics Department has propagated gradebooks with appropriate weights.66

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ResourcesEngage NYhttp://www.engageny.org/video-library?f[0]=im_field_subject%3A19Common Core Toolshttp://commoncoretools.me/http://www.ccsstoolbox.com/http://www.achievethecore.org/steal-these-toolsAchieve the Corehttp://achievethecore.org/dashboard/300/search/6/1/0/1/2/3/4/5/6/7/8/9/10/11/12Manipulativeshttp://nlvm.usu.edu/en/nav/vlibrary.htmlhttp://www.explorelearning.com/index.cfm?method=cResource.dspBrowseCorrelations&v=s&id=USA-000http://www.thinkingblocks.com/Illustrative Math Project :http://illustrativemathematics.org/standards/k8 Inside Mathematics: http://www.insidemathematics.org/index.php/tools-for-teachersSample Balance Math Tasks: http://www.nottingham.ac.uk/~ttzedweb/MARS/tasks/Georgia Department of Education:https://www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspxGates Foundations Tasks:http://www.gatesfoundation.org/college-ready-education/Documents/supporting-instruction-cards-math.pdfMinnesota STEM Teachers’ Center: http://www.scimathmn.org/stemtc/frameworks/721-proportional-relationshipsSingapore Math Tests K-12: http://www.misskoh.comMobymax.com: http://www.mobymax.com

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