6th Grade Mathematics - Orange Board of Education · Web view1st Grade Mathematics. Addition and Subtraction with Regrouping. ... Write a key word or phrase from the task on the board.

1st Grade MathematicsAddition and Subtraction with RegroupingAddition and Subtraction StrategiesUnit IV Curriculum Map: April 6th, 2017-June 19th, 2017

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ORANGE PUBLIC SCHOOLSOFFICE OF CURRICULUM AND INSTRUCTIONOFFICE OF MATHEMATICS

First Grade Unit IV 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. ELL & SPED Considerations p. 16X. K-2 Concept Map p. 21XI. First Grade Unit IV NJSLS p. 22XII. Eight Mathematical Practices p. 40XIII. Ideal Math Block p. 42XIX. Math In Focus Lesson Structure p. 46XX. Ideal Math Block Planning Template p. 47XXI. Planning Calendar p. 52XXII. Instructional and Assessment Framework p. 54XXIII. Data Driven Instruction p. 58XXI Data Analysis Form p. 59

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V.XXV. Resources p. 64Office 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

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mathematics (2) set course expectations for each of our courses of study and (3) encourage 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:1.OA.1Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.1.OA.6Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).Learning Goal(s):Students will use multiple representations to solve multi-step addition and/or subtraction situations (1.OA.1) and explain the connection between various solution paths (1.OA.6).Student Friendly Version:

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We are learning to represent and solve word problems and explain how different representations match the story situation and the math operations.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. 

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 Draw or Show I am not able to draw or show my thinking.

I can draw, but not show my thinking;orI can show but not draw my thinking;I can draw and show my thinking I can draw, show and talk about 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.19

K-2 Math Fact Fluency ExpectationK.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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English Language Learners (ELL) and Special Education (SPED) Considerations In order to develop proficiency in the Standard for Mathematical Practice 3 (Construct Viable Arguments and Critique the Reasoning of Others) and Standard for Mathematical Practice 4 (Model with Mathematics), it is important to provide English Language Learners (ELLs) and Special Education Students with two levels of access to the tasks: language access and content access.Language AccessIn the tasks presented, we can distinguish between the vocabulary and the language functions needed to provide entry points to the math content. These vocabulary words and language functions must be explicitly taught to ensure comprehension of the tasks. Some ways this can be done are by using the following approaches:1. Introduce the most essential vocabulary/language functions before beginning the tasks. Select words and concepts that are essential in each task.

Vocabulary Words: Tier I (Nonacademic language) Mostly social language; terms used regularly in everyday situations (e.g., small, orange, clock) Tier II (General academic language) Mostly academic language used regularly in school but not directly associated with mathematics (e.g., combine, describe, consequently), and academic language broadly associated with mathematics (e.g., number, angle, equation, average, product) Tier III (Math technical language) Academic language associated with specific math topics (e.g., perfect numbers, supplementary angles, quadratic equations, mode, median)

Language Functions: Pronounce each word for students and have them repeat after you. Introduce the vocabulary in a familiar and meaningful context and then again in a contentspecific setting. Math-specific examples include but are not limited to the following: explain, describe, inform, order, classify, analyze, infer, solve problems, define, generalize, interpret, hypothesize.

2. Use visuals when introducing new words and concepts. Provide experiences that help demonstrate the meaning of the vocabulary words (e.g., realia, pictures, photographs, and graphic organizers). Write key words on the board, and add gestures to help students interpret meaning. Have students create their own visuals to aid in their learning. For example, assign a few content-specific vocabulary words to each student, and ask them to write student-friendly definitions and draw pictures to show what the words mean.

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3. Build background knowledge. Explicit links to previously taught lessons, tasks, or texts should be emphasized to activate prior knowledge. Review relevant vocabulary that has already been introduced, and highlight familiar words that have a new meaning. Access the knowledge that students bring from their native cultures.

4. Promote oral language development through cooperative learning groups. ELLs need ample opportunities to speak English and authentic reasons to use academic language. Working in small groups is especially beneficial because ELLs learn to negotiate the meanings of vocabulary words with their classmates.

5. Native Language Support Full proficiency in the native language leads to higher academic gains in English. Because general structural and functional characteristics of languages transfer, allowing second language learners to access content in the native language provides them with a way to construct meaning in English. In order to assist ELLs, the strategic use of the native language can be incorporated into English instruction as a support structure in order to clarify, build prior knowledge, extend comprehension, and bridge experiences. This can be integrated into a teacher’s instructional practices through technology, human resources (e.g., paraprofessionals, peers, and parents), native language materials, and flexible grouping.

6. Possible SentencesMoore, D.W., & Moore, S.A. (1986). "Possible sentences." In Reading in the content areas: Improving classroom instruction. Dubuque, IA: Kendall/Hunt. Possible Sentences is a pre-reading strategy that focuses on vocabulary building and student prediction prior to reading. In this strategy, teachers write the key words and phrases of a selected text on the chalkboard. Students are asked to:• Define all of the terms• Group the terms into related pairs• Write sentences using these word pairsSteps to Possible Sentences1) Prior to the reading assignment, list all essential vocabulary words in the task on the board.2) Working in pairs, ask students to define the words and select pairs of related words from the list.3) Ask students to write sentences using each of the word pairs that they might expect to appear in the task, given its title and topic.4) Select several students to write their possible sentences on the board.5) Engage the students in a discussion of the appropriateness of the word pairing and the plausibility of each sentence as a possible sentence in the selection.

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6) Have students read the task and test the accuracy of their predictions. Sentences that are not accurate should be revised.7) Poll the class for common accurate and inaccurate predictions. Discuss possible explanations for the success or failure of these predictions.8) Introduce students to sentence frames which reinforce sentence structure while enabling ELLs to participate in classroom and/or group discussion.7. The Frayer ModelFrayer, D., Frederick, W. C., and Klausmeier, H. J. (1969). A Schema for testing the level of cognitive mastery. Madison, WI: Wisconsin Center for Education Research.The Frayer Model is a graphic organizer used for word analysis and vocabulary building. It assists students in thinking about and describing the meaning of a word or concept by:• Defining the term• Describing its essential characteristics• Providing examples of the idea• Offering non-examples of the ideaSteps to the Frayer Model1) Explain the Frayer Model graphic organizer to the class. Use a common word to demonstrate the various components of the form. Model the type and quality of desired answers when giving this example.2) Select a list of key concepts from the task. Write this list on the chalkboard and review it with the class before students read the task.3) Divide the class into student pairs. Assign each pair one of the key concepts and have them read the task carefully to define this concept. Have these groups complete the four-square organizer for this concept.4) Ask the student pairs to share their conclusions with the entire class. Use these presentations to review the entire list of key concepts.

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8. Semantic WebbingMaddux, C. D., Johnson, D. L., & Willis, J. W. (1997). Educational computing: Learning with tomorrow's technologies. Boston: Allyn & Bacon.Semantic Webbing builds a graphical representation of students' knowledge and perspectives about the key themes of a task before and after the learning experience. Semantic Webbing achieves three goals:

• Activating students' prior knowledge and experience• Helping students organize both their prior knowledge and new information• Allowing students to discover relationships between their prior and new knowledgeSteps to Semantic Webbing1) Write a key word or phrase from the task on the board. 2) Have students think of as many words as they know that relate to this key idea. Write these words on the side on the chalkboard.3) Ask students to group these words into logical categories and label each category with a descriptive title.4) Encourage students to discuss/debate the choice of the category for each word.5) Write the students' conclusions (the categories and their component words) on the chalkboard.6) Have the students read the task in pairs and repeat the process above.7) When they finish reading, have students add new words and categories related to the key idea.Native Language Support: Full proficiency in the native language leads to higher academic gains in English. Because general structural and functional characteristics of language transfer, allowing second language learners to access content in the native language provides them with a way to construct meaning in English. In order to assist ELLs, the strategic use of the native language can be incorporated into English instruction as a support structure to clarify, to build prior knowledge, to extend comprehension, and to bridge experiences. This can be integrated into a teacher’s instructional practice through the following: technology, human resources (e.g., paraprofessionals, peers, and parents), native language materials, and flexible grouping.Content AccessWhen engaging ELL/SPED students in cognitively demanding tasks, teachers should consider which concepts the ELLs/SPEDs are likely to encounter when accessing mathematics and which of these pose the most challenges. Teachers should consider what the student is required to know as well as be able to do.What is the mathematics in the task? What prior knowledge is required in order for ELL/SPED students to proceed? In order to activate prior knowledge and prepare ELL/SPED for the demands of the tasks in the lesson, we suggest that they engage in a different but similar task prior to working on the selected performance assessment tasks, such as the following:

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1. Use of ManipulativesProvide ELL/SPED students with manipulatives when appropriate. While there are different types of manipulatives available commercially, teacher-made materials are recommended and encouraged. Manipulatives are always appropriate when introducing a concept regardless of the grade.2. Graphic OrganizersGraphic organizers, such as Venn diagrams, Frayer Models, charts and/or tables, help ELLs/SPEDs understand relationships, recognize common attributes, and make associations with the concepts being discussed.3. Use of TechnologyTechnology must be integrated whenever possible. Various software and internet-based programs can also be very beneficial, many of which are available in the ELLs’ native languages. Use of technology develops and reinforces basic skills.4. Differentiated InstructionWhile all students can benefit from differentiated instruction, it is crucial for teachers to identify the different learning modalities of their ELLs/SPEDs. Teachers and ELLs/SPEDs are collaborators in the learning process. Teachers must adjust content, process, and product in response to the readiness, interests, and learning profiles of their students. In order to create and promote the appropriate climate for ELLs/SPEDs to succeed, teachers need to know, engage, and assess the learner.

5. Assessment for Learning (AfL)Whenever ELL/SPED students are engaged in tasks for the purpose of formative assessments, the strategies of Assessment for Learning (AfL) are highly recommended. AfL consists of five key strategies for effective formative assessment:

1) Clarifying, sharing and understanding goals for learning and criteria for success with learners2) Engineer effective classroom discussions, questions, activities, and tasks that elicit evidence of students’ learning3) Providing feedback that moves learning forward4) Activating students as owners of their own learning5) Activating students as learning resources for one anotherScaffolding: A Tool to AccessibilityIn order to be successful members of a rigorous academic environment, ELLs/SPED need scaffolds to help them access curriculum. These scaffolds are temporary, and the process of

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constructing them and then removing them when they are no longer needed is what makes them a valuable tool in the education of ELLs/SPEDs. The original definition of scaffolding comes from Jerome Bruner (1983), who defines scaffolding as “a process of setting up the situation to make the child’s entry easy and successful, and then gradually pulling back and handing the role to the child as he becomes skilled enough to manage it.” The scaffolds are placed purposefully to teach specific skills and language. Once students learn these skills and gain the needed linguistic and content knowledge, these scaffolds are no longer needed. Nevertheless, each child moves along his/her own continuum, and while one child may no longer need the scaffolds, some students may still depend on them. Thus, constant evaluation of the process is an inevitablestep in assuring that scaffolds are ujsed successfully.The scaffolding types necessary for ELLs/SPEDs are modeling, activating and bridging prior knowledge and/or experiences, text representation, metacognitive development, contextualization, and building schema: • Modeling: finished products of prior students’ work, teacher-created samples, sentence starters, writing frameworks, shared writing, etc.

• Activating and bridging prior knowledge and/or experiences: using graphic organizers, such as anticipatory guides, extended anticipatory guide, semantic maps, interviews, picture walk discussion protocols, think-pair-share, KWL, etc.• Text representation: transforming a piece of writing into a pictorial representation, changing one genre into another, etc.• Metacognitive development: self-assessment, think-aloud, asking clarifying questions, using a rubric for self evaluation, etc.• Contextualization: metaphors, realia, pictures, audio and video clips, newspapers, magazines, etc.• Building schema: bridging prior knowledge and experience to new concepts and ideas, etc.

NYC Department of Eduction, ELL Considerations for Common Core-Aligned Tasks in Mathematicshttp://schools.nyc.gov/NR/rdonlyres/9E62A2F2-4C5C-4534-968B-5487A7BD3742/0/GeneralMathStrategiesforELLs_082811.pdfRetrieved on December 5, 2016

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K-2 CONCEPT MAP

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First Grade Unit IV In this Unit Students will:1.OA.1-8

Solve addition and subtraction situations involving: Adding to, Taking From , Putting Together, Taking Apart, and Comparing situations.

Apply the following problem solving strategies Use of objects and/or drawings Counting On Making Ten Decomposing Numbers Properties of Operations Relationship between Addition and Subtraction

1.NBT.1-4 Count to 120 starting at any number less than 120 Reade and write numerals Represent number of objects with a written numeral Understand the value of a digit within a number Compare two-digit numbers

Mathematical Practices Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate mathematical tools. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.

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New Jersey Student Learning Standards: Operations and Algebraic Thinking

1.OA.1Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

First grade students extend their experiences in Kindergarten by working with numbers to 20 to solve a new type of problem situation: Compare (See Table 1 at end of document for examples of all problem types). In a Compare situation, two amounts are compared to find “How many more” or “How many less”.

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New Jersey Student Learning Standards: Operations and Algebraic Thinking1.OA.2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20.

First Grade students solve multi-step word problems by adding (joining) three numbers whose sum is less than or equal to 20, using a variety of mathematical representations.Example: Mrs. Smith has 4 oatmeal raisin cookies, 5 chocolate chip cookies, and 6 gingerbread cookies. How many cookies does Mrs. Smith have?Student A:I put 4 counters on the Ten Frame for the oatmeal raisin cookies. Then, I put 5 different color counters on the ten frame for the chocolate chip cookies. Then, I put another 6 color counters out for the gingerbread cookies. Only one of the gingerbread cookies fit, so I had 5 leftover. Ten and five more makes 15 cookies. Mrs. Smith has 15 cookies.

Student B:I used a number line. First I jumped to 4, and then I jumped 5 more. That’s 9. I broke up 6 into 1 and 5 so I could jump 1 to make 10. Then, I jumped 5 more and got 15. Mrs. Smith has 15 cookies.

Student C:I wrote: 4 + 5 + 6 = . I know that 4 and 6 equals 10, so the oatmeal raisin and gingerbread equals 10 cookies.32

Then I added the 5 chocolate chip cookies. 10 and 5 is 15. So, Mrs. Smith has 15 cookies.

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New Jersey Student Learning Standards: Operations and Algebraic Thinking1.OA.3 Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) Students need not use formal terms for these properties.Elementary students often believe that there are hundreds of isolated addition and subtraction facts to be mastered. However, when students understand the commutative and associative properties, they are able to use relationships between and among numbers to solve problems. First Grade students apply properties of operations as strategies to add and subtract. Students do not use the formal terms “commutative” and associative”. Rather, they use the understandings of the commutative and associative property to solve problems.

Students use mathematical tools and representations (e.g., cubes, counters, number balance, number line, 100 chart) to model these ideas.Commutative Property Examples: CubesA student uses 2 colors of cubes to make as many different combinations of 8 as possible. When recording the combinations, the student records that 3 green cubes and 5 blue cubes

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equals 8 cubes in all. In addition, the student notices that 5 green cubes and 3 blue cubes also equals 8 cubes.Associative Property Examples:Number Line: = 5 + 4 + 5Student A: First I jumped to 5. Then, I jumped 4 more, so I landed on 9. Then I jumped 5 more and landed on 14.

Student B: I got 14, too, but I did it a different way. First I jumped to 5. Then, I jumped 5 again. That’s 10.Then, I jumped 4 more. See, 14!

Mental Math: There are 9 red jelly beans, 7 green jelly beans, and 3 black jelly beans. How many jelly beans are there in all?Student: “I know that 7 + 3 is 10. And 10 and 9 is 19. There are 19 jelly beans.”1.OA.4 Understand subtraction as an unknown-addend problem

First Graders often find subtraction facts more difficult to learn than addition facts. By understanding the relationship between addition and subtraction, First Graders are able to use various strategies described below to solve subtraction problems.For Sums to 10*Think-Addition:Think-Addition uses known addition facts to solve for the unknown part or quantity within a problem. When

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students use this strategy, they think, “What goes with this part to make the total?” The think-addition strategy is particularly helpful for subtraction facts with sums of 10 or less and can be used for sixty-four of the 100 subtraction facts. Therefore, in order for think-addition to be an effective strategy, students must have mastered addition facts first.

For example, when working with the problem 9 - 5 = , First Graders think “Five and what makes nine?”, rather than relying on a counting approach in which the student counts 9, counts off 5, and then counts what’s left. When subtraction is presented in a way that encourages students to think using addition, they use known addition facts to solve a problem.Example: 10 - 2 = Student: “2 and what make 10? I know that 8 and 2 make 10. So, 10 - 2 = 8.”For Sums Greater than 10The 36 facts that have sums greater than 10 are often considered the most difficult for students to master. Many students will solve these particular facts with Think-Addition (described above), while other students may use other strategies described below, depending on the fact. Regardless of the strategy used, all strategies focus on the relationship between addition and subtraction and often use 10 as a benchmark number.*Build Up Through 10:This strategy is particularly helpful when one of the numbers to be subtracted is 8 or 9. Using 10 as a bridge, either 1 or 2 are added to make 10, and then the remaining amount is added for the final sum.Example: 15 -9 = Student A: “I’ll start with 9. I need one more to make 10. Then, I need 5 more to make 15. That’s 1 and 5- so it’s 6. 15 0 9 = 6.”Student B: “I put 9 counters on the 10 frame. Just looking at it I can tell that I need 1 more to get to 10. Then I need 5 more to get to 15. So, I need 6 counters.”

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*Back Down Through 10This strategy uses take-away and 10 as a bridge. Students take away an amount to make 10, and then take away the rest. It is helpful for facts where the ones digit of the two-digit number is close to the number being subtracted.Example: 16 – 7 = Student A: “I’ll start with 16 and take off 6. That makes 10. I’ll take one more off and that makes 9. 16 – 7 = 9.”Student B: “I used 16 counters to fill one ten frame completely and most of the other one. Then, I can take these 6 off from the 2nd ten frame. Then, I’ll take one more from the first ten frame. That leaves 9 on the ten frame.”

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1.OA.5 Relate counting to addition and subtractionWhen solving addition and subtraction problems to 20, First Graders often use counting strategies, such as counting all, counting on, and counting back, before fully developing the essential strategy of using 10 as a benchmark number. Once students have developed counting strategies to solve addition and subtraction problems, it is very important to move students toward strategies that focus on composing and decomposing number using ten as a benchmark number, as discussed in 1.OA.6, particularly since counting becomes a hindrance when working with larger numbers. By the end of First Grade, students are expected to use the strategy of 10 to solve problems.Counting All: Students count all objects to determine the total amount.Counting On & Counting Back: Students hold a “start number” in their head and count on/back from that number.Example: 15 + 2 =

Example: 12 – 3 =

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1.OA.6Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

In First Grade, students learn about and use various strategies to solve addition and subtraction problems. When students repeatedly use strategies that make sense to them, they internalize facts and develop fluency for addition and subtraction within 10. When students are able to demonstrate fluency within 10, they are accurate, efficient, and flexible. First Graders then apply similar strategies for solving problems within 20, building the foundation for fluency to 20 in Second Grade.Developing Fluency for Addition & Subtraction within 10Example: Two frogs were sitting on a log. 6 more frogs hopped there. How many frogs are sitting on the log now?

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Add and Subtract within 20Example: Sam has 8 red marbles and 7 green marbles. How many marbles does Sam have in all?

Example: There were 14 birds in the tree. 6 flew away. How many birds are in the tree now?

1.OA.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.40

In order to determine whether an equation is true or false, First Grade students must first understand the meaning of the equal sign. This is developed as students in Kindergarten and First Grade solve numerous joining and separating situations with mathematical tools, rather than symbols. Once the concepts of joining, separating, and “the same amount/quantity as” are developed concretely, First Graders are ready to connect these experiences to the corresponding symbols (+, -, =). Thus, students learn that the equal sign does not mean “the answer comes next”, but that the symbol signifies an equivalent relationship that the left side ‘has the same value as’ the right side of the equation.When students understand that an equation needs to “balance”, with equal quantities on both sides of the equal sign, they understand various representations of equations, such as:• an operation on the left side of the equal sign and the answer on the right side (5 + 8 = 13)• an operation on the right side of the equal sign and the answer on the left side (13 = 5 + 8)• numbers on both sides of the equal sign (6 = 6)• operations on both sides of the equal sign (5 + 2 = 4 + 3).Once students understand the meaning of the equal sign, they are able to determine if an equation is true (9 = 9) or false (9 = 8).

1.OA.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ – 3, 6 + 6 = _.First Graders use their understanding of and strategies related to addition and subtraction as described in 1.OA.4 and 1.OA.6 to solve equations with an unknown. Rather than symbols, the unknown symbols are boxes or pictures.Example: Five cookies were on the table. I ate some cookies. Then there were 3 cookies. How many cookies did I eat?Student A: What goes with 3 to make 5? 3 and 2 is 5. So, 2 cookies were eaten.Student B: Fiiivee, four, three (holding up 1 finger for each count). 2 cookies were eaten (showing 2 fingers).Student C: We ended with 3 cookies. Threeeee, four, five (holding up 1 finger for each count). 2 cookies were eaten (showing 2 fingers).Example: Determine the unknown number that makes the equation true. 5 - = 2Student: 5 minus something is the same amount as 2. Hmmm. 2 and what makes 5? 3! So, 5 minus 3 equals 2.

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Now it’s true!New Jersey Student Learning Standards: Numbers and Operation in Base Ten

1.NBT.1 Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count on from a number ending at any number up to 120. Recognize and explain patterns with numerals on a hundreds chart. Understand that the place of a digit determines its value. For example, students recognize that 24 is different from and less than 42.) Explain their thinking with a variety of examples. Read and write numerals to 120.

Students extend the range of counting numbers, focusing on the patterns evident in written numerals. This is the foundation for thinking about place value and the meaning of the digits in a numeral. Students are also expected to read and write numerals to 120.

1.NBT.2

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as Special cases:a. 10 can be thought of as a bundle of ten ones — called a “ten.”b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

a. 10 can be thought of as a bundle of ten ones — called a “ten.”

First Grade students are introduced to the idea that a bundle of ten ones is called “a ten”. This is known as unitizing. When First Grade students unitize a group of ten ones as a whole unit (“a ten”), they are able to count groups as though they were individual objects. For example, 4 trains of ten cubes each have a value of 10 and would be counted as 40 rather than as 4. This is a monumental shift in thinking, and can often be challenging for young children to consider a group of something as “one” when all previous experiences have been counting single objects. This is the foundation of the place value system and requires time and rich experiences with concrete manipulatives to develop.

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A student’s ability to conserve number is an important aspect of this standard. It is not obvious to young children that 42 cubes is the same amount as 4 tens and 2 left-overs. It is also not obvious that 42 could also be composed of 2 groups of 10 and 22 leftovers. Therefore, first graders require ample time grouping proportional objects (e.g., cubes, beans, beads, ten-frames) to make groups of ten, rather than using pre-grouped materials (e.g., base tenblocks, pre-made bean sticks) that have to be “traded” or are non-proportional (e.g., money).

Example: 42 cubes can be grouped many different ways and still remain a total of 42 cubes.

“We want children to construct the idea that all of these are the same and that the sameness is clearly evident byvirtue of the groupings of ten. Groupings by tens is not just a rule that is followed but that any grouping by tens,

including all or some of the singles, can help tell how many.” (Van de Walle & Lovin, p. 124)

b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

First Grade students extend their work from Kindergarten when they composed and decomposed numbers from 11 to 19 into ten ones and some further ones. In Kindergarten, everything was thought of as individual units: “ones”. In First Grade, students are asked to unitize those ten individual ones as a whole unit: “one ten”. Students in first grade explore the idea that the teen numbers (11 to 19) can be expressed as one ten and some leftover ones. Ample experiences with a variety of groupable materials that are proportional (e.g., cubes, links, beans, beads) and ten frames help students develop this concept.

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Example:Here is a pile of 12 cubes. Do you have enough to make a ten? Would you have any leftover? If so, how many leftovers would you have?

Student AI filled a ten frame to make one ten and had two counters left over. I had enough to make a ten with some leftover. The number 12 has 1 ten and 2 ones.

Student BI counted out 12 cubes. I had enough to make 10. I now have 1 ten and 2 cubes left over. So the number 12 has 1 ten and 2 ones.In addition, when learning about forming groups of 10, First Grade students learn that a numeral can stand for many different amounts, depending on its position or place in a number.

Example: Comparing 19 to 91

c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

First Grade students apply their understanding of groups of ten as stated in 1.NBT.2b to decade numbers (e.g. 10, 20, 30, 40). As they work with groupable objects, first grade students understand that 10, 20, 30…80, 90 are comprised of a certain amount of groups of tens with none left-over.

1.NBT.3Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

First Grade students use their understanding of groups and order of digits to compare two numbers by examining the amount of tens and ones in each number. After numerous experiences verbally comparing two sets of objects using comparison vocabulary (e.g., 42 is more than 31. 23 is less than 52, 61 is the same amount as 61.), first grade students connect the vocabulary to the symbols: greater than (>), less than (<), equal to (=).

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Example: Compare these two numbers. 42 __ 45

1.NBT.4

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.First Grade students use concrete materials, models, drawings and place value strategies to add within 100. They do so by being flexible with numbers as they use the base-ten system to solve problems. The standard algorithm of carrying or borrowing is neither an expectation nor a focus in First Grade. Students use strategies for addition and subtraction in Grades K-3. By the end of Third Grade students use a range of algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction to fluently add and subtract within 1000. Students are expected to fluently add and subtract multi-digit whole numbers using the standard algorithm by the end of Grade 4.Example: 24 red apples and 8 green apples are on the table. How many apples are on the table?Student A:I used ten frames. I put 24 chips on 3 ten frames. Then, I counted out 8 more chips. 6 of them filled up the third ten-frame.That meant I had 2 left over. 3 tens and 2 left over. That’s 32. So, there are 32 apples on the table.

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Student B:I used an open number line. I started at 24. I knew that I needed 6 more jumps to get to 30. So, I broke apart 8 into 6 and 2. I took 6 jumps to land on 30 and then 2 more. I landed on 32. So, there are 32 apples on the table.

Student C:

I turned 8 into 10 by adding 2 because it’s easier to add.So, 24 and ten more is 34. But, since I added 2 extra, I had to take them off again.34 minus 2 is 32. There are 32 apples on the table.

Example: 63 apples are in the basket. Mary put 20 more apples in the basket. How many apples are in the basket?Student A:I used ten frames. I picked out 6 filled ten frames. That’s 60. I got the ten frame with 3 on it. That’s 63. Then, I picked one more filled ten frame for part of the 20 that Mary put in. That made 73. Then, I got one more filled ten frame to make the rest of the 20 apples from Mary. That’s 83. So, there are 83 apples in the basket.

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Student B:I used a hundreds chart. I started at 63 and jumped down one row to 73. That means I moved 10 spaces. Then, I jumped down one more row (that’s another 10 spaces) and landed on 83. So, there are 83 apples in the basket.

Student C”I knew that 10 more than 63 is 73. And 10 more than 73 is 83. So, there are 83 apples in the basket.

1.MD.4 Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

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First Grade students collect and use categorical data (e.g., eye color, shoe size, age) to answer a question. The data collected are often organized in a chart or table. Once the data are collected, First Graders interpret the data to determine the answer to the question posed. They also describe the data noting particular aspects such as the total number of answers, which category had the most/least responses, and interesting differences/similarities between the categories.

As the teacher provides numerous opportunities for students to create questions, determine up to 3 categories of possible responses, collect data, organize data, and interpret the results, First Graders build a solid foundation for future data representations (picture and bar graphs) in Second Grade.

Example:

Survey Station During Literacy Block, a group of students work at the Survey Station. Each student writes a question, creates up to 3 possible answers, and walks around the room collecting data from classmates. Each student then interprets the data and writes 2-4 sentences describing the results. When all of the students in the Survey Station have completed their own data collection, they each share with one another what they discovered. They ask clarifying questions of one another regarding the data, and make revisions as needed. They later share their results with the whole class.

Student:

The question, “What is your favorite flavor of ice cream?” is posed and recorded. The categories chocolate, vanilla and strawberry are determined as anticipated responses and written down on the recording sheet. When asking each classmate about their favorite flavor, the student’s name is written in the appropriate category. Once the data are collected, the student counts up the amounts for each category and records the amount. The student then analyzes the data by carefully looking at the data and writes 4 sentences about the data.

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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. 

1

Make sense of problems and persevere in solving them

Mathematically proficient students in Second Grade examine problems and tasks, can make sense of the meaning of the task and find an entry point or a way to start the task. Second Grade students also develop a foundation for problem solving strategies and become independently proficient on using those strategies to solve new tasks. In Second Grade, students’ work continues to use concrete manipulatives and pictorial representations as well as mental mathematics. Second Grade 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, mathematically proficient students complete a task by asking themselves the question, “Does my answer make sense?”

2 Reason abstractly and quantitatively

Mathematically proficient students in Second Grade make sense of quantities and relationships while solving tasks. This involves two processes- decontextualizing and contextualizing. In Second Grade, students represent situations by decontextualizing tasks into numbers and symbols. For example, in the task, “There are 25 children in the cafeteria and they are joined by 17 more children. How many students are in the cafeteria? ” Second Grade students translate that situation into an equation, such as: 25 + 17 = __ and then solve the problem. Students also contextualize situations during the problem solving process. For example, while solving the task above, students can refer to the context of the task to determine that they need to subtract 19 since 19 children leave. The processes of reasoning also other areas of 49

mathematics such as determining the length of quantities when measuring with standard units.

3Construct viable arguments and critique the reasoning of others

Mathematically proficient students in Second Grade accurately use definitions and previously established solutions to construct viable arguments about mathematics. During discussions about problem solving strategies, students constructively critique the strategies and reasoning of their classmates. For example, while solving 74 - 18, students may use a variety of strategies, and after working on the task, can discuss and critique each other’s reasoning and strategies, citing similarities and differences between strategies.

4

Model with mathematics

Mathematically proficient students in Second Grade 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. Second Grade students use concrete manipulatives and pictorial representations to provide further explanation of the equation. Likewise, Second Grade students are able to create an appropriate problem situation from an equation. For example, students are expected to create a story problem for the equation 43 + 17 = ___ such as “There were 43 gumballs in the machine. Tom poured in 17 more gumballs. How many gumballs are now in the machine?”

5 Use appropriate tools strategicallyMathematically proficient students in Second Grade have access to and use tools appropriately. These tools may include snap cubes, place value (base ten) blocks, hundreds number boards, number lines, rulers, and concrete geometric shapes (e.g., pattern blocks, 3-d solids). Students also have experiences with educational technologies, such as calculators and virtual manipulatives, which support conceptual understanding and higher-order thinking skills. During classroom instruction, students have access to various mathematical tools as well as paper, and determine which tools are the most appropriate to use. For example, while measuring the length of the hallway, students can explain why a yardstick is more appropriate

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to use than a ruler.

6Attend to precisionMathematically proficient students in Second Grade are precise in their communication, calculations, and measurements.In all mathematical tasks, students in Second Grade communicate clearly, using grade-level appropriate vocabulary accurately as well as giving precise explanations and reasoning regarding their process of finding solutions. For example, while measuring an object, care is taken to line up the tool correctly in order to get an accurate measurement. During tasks involving number sense, students consider if their answer is reasonable and check their work to ensure the accuracy of solutions.

7Look for and make use of structureMathematically proficient students in Second Grade carefully look for patterns and structures in the number system and other areas of mathematics. For example, students notice number patterns within the tens place as they connect skip count by 10s off the decade to the corresponding numbers on a 100s chart. While working in the Numbers in Base Ten domain, students work with the idea that 10 ones equal a ten, and 10 tens equals 1 hundred. In addition, Second Grade students also make use of structure when they work with subtraction as missing addend problems, such as 50- 33 = __ can be written as 33+ __ = 50 and can be thought of as,” How much more do I need to add to 33 to get to 50?”

8Look for and express regularity in repeated reasoningMathematically proficient students in Second Grade begin to look for regularity in problem structures when solving mathematical tasks. For example, after solving two digit addition problems by decomposing numbers (33+ 25 = 30 + 20 + 3 +5), students may begin to generalize and frequently apply that strategy independently on future tasks. Further, students begin to look for strategies to be more efficient in computations, including doubles strategies and making a ten. Lastly, while solving all tasks, Second Grade students accurately check for the reasonableness of their solutions during and after completing the task.

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LEARN: Whole GroupAnchor Task: Math In Focus Learn

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

INDEPENDENT PRACTICE: IndividualMath In Focus Let’s Practice, Workbook, Reteach, Extra Practice, Enrichment

MATH WORKSTATIONS: 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

1st & 2nd Grade Ideal Math Block Essential Components

FLUENCY: Partner/Small GroupCONCRETE, PICTORIAL, and ABSTRACT approaches to support ARITHMETIC

FLUENCY and FLUENT USE OF STRATEGIES.

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

EXIT TICKET (DOL): IndividualStudents complete independently; Used to guide instructional decisions;

Used to set instructional goals for students;

5 min.

15-20 min.

5 min.

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.

15-20 min.

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 5

56

6 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

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:1.OA.61.NBT.1Strategy:

Tool(s):

Mat

h In

Foc

us/E

nGag

eNY

Launch

Exploration

Independent Practice

61

Diffe

rent

iatio

n: M

ath

Wor

ksta

tions

Small Group Instruction

Tech. Lab

Problem Solving Lab

CCSS:1.OA.11.OA.21.NBT.4

Fluency Lab1.OA.61.NBT.51.NBT.6

Strategy:

Tool(s):

Math Journal

MP3: Construct viable arguments and critique the reasoning of othersSummary

Exit Ticket

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Danielson Framework for Teaching: Domain 1: Planning and Preparation Lesson 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 LabComponent 1C: Setting Instructional Outcomes

Lesson plan objectives are aligned to one or more New Jersey Student Standards for LearningConnections made to previous learningOutcomes: student artifacts are differentiatedComponent 1D: Demonstrating Knowledge of Resources

District 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 UNDERSTANDINGLesson Plans show evidences of CONCRETE, PICTORIAL, and ABSTRACT representationAlignment between OBJECTIVES, APPLICATION, and ASSESSMENT evident

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___________________________________________________________________________________________________________________________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 April 2017

Monday Tuesday Wednesday Thursday Friday3 4 5 6 END OF MP 3 7

10 11 12 13 14

17 18 19 20 21

24 25 26 27MIF Ch. 14 Test PrepMIF Ch. 14 Performance Task

28

May 2017Monday Tuesday Wednesday Thursday Friday

1 2 3 4 58 9 10 11

MIF Ch. 16 Test PrepMIF Ch. 16 Performance Task

12

15 16 17 18 1922 23 24

MIF Ch. 17 Test Prep25 26

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DISTRICT UNIT III ASSESSMENT

SPRING RECESS

MIF Ch. 17 Performance Task29 30 31Planning Calendar June 2017

Monday Tuesday Wednesday Thursday Friday1 2

5 6MIF Ch. 11 Test PrepMIF Ch. 11 Performance Task

7 8 9

12 13 14 15 16

19 END OF MP20 21 22 23

26 27 28 29 30

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DISTRICT END OF YEAR ASSESSMENT

First Grade Unit IV Instructional and Assessment FrameworkRecommended Activities CCSS Notes April 10th- , 2017 Math In Focus Ch. 14 OpenerRecall Prior KnowledgeCh. 14 Pre-Test

1.OA.31.OA.41.OA.61.OA.81.NBT.5

April 11th- , 2017 Math In Focus Ch. 14 Lesson 1 Mental Addition

April 12th- , 2017 Math In Focus Ch. 14 Lesson 1Mental Addition

April 13th- , 2017 Math In Focus Ch. 14 Lesson 2Mental SubtractionApril 14th – 21st, 2017 Spring RecessApril 24th , 2017 Math In Focus Ch. 14 Lesson 2Mental SubtractionApril 25th , 2017 Math In Focus Ch. 14 Put On Your Thinking CapApril 26th , 2017 Math In Focus Ch. 14 Chapter Wrap UpApril 27th , 2017 Math In Focus Ch. 14 Test PrepMath In Focus Ch. 14 Performance TaskApril 28th , 2017 Math In Focus Ch. 16 OpenerRecall Prior KnowledgeCh. 16 Pre-Test

1.OA.31.OA.41.OA.61.OA.81.NBT.5May 1st , 2017 Math In Focus Ch. 16 Lesson 1

Counting to 120May 2nd, 2017 Math In Focus Ch. 16 Lesson 1

Counting to 120May 3rd, 2017 Math In Focus Ch. 16 Lesson 2Place ValueMay 4th , 2017 Math In Focus Ch. 16 Lesson 3Comparing, Ordering, PatternsMay 5th , 2017 Math In Focus Ch. 16 Lesson 3Comparing, Ordering, Patterns

Math In Focus Ch. 16 Lesson 368

May 8th, 2017 Comparing, Ordering, PatternsMay 9th, 2017 Math In Focus Ch. 16Put On Your Thinking CapMay 10th, 2017 Math In Focus Ch. 16 Chapter Wrap UpMay 11th, 2017 Math In Focus Ch. 16 Test PrepMath In Focus Ch. 16 Performance TaskMay 12th, 2017 Math In Focus Ch. 17 OpenerRecall Prior KnowledgeCh. 17 Pre-Test 1.NBT.21.NBT.41.NBT.6

1.OA.4May 15th, 2017 Math In Focus Ch. 17 Lesson 1

Addition Without RegroupingMay 16th, 2017 Math In Focus Ch. 17 Lesson 2

Addition With RegroupingMay 17th, 2017 Math In Focus Ch. 17 Lesson 2

Addition With RegroupingMay 18th, 2017 Math In Focus Ch. 17 Lesson 3Subtraction Without RegroupingMay 19th, 2017 Math In Focus Ch. 17 Lesson 4Subtraction With RegroupingMay 22nd, 2017 Math In Focus Ch. 17Put On Your Thinking CapMay 23rd, 2017 Math In Focus Ch. 17 Chapter Wrap UpMay 24th, 2017 Math In Focus Ch. 17 Test PrepMath In Focus Ch. 17 Performance TaskMay 25th, 2017 Math In Focus Ch. 11 OpenerRecall Prior KnowledgeCh. 11 Pre-Test

1.MD.41.OA.8May 26th, 2017 Math In Focus Ch. 11 Lesson 1

Simple Picture Graphs

May 29th, 2017 Math In Focus Ch. 11 Lesson 1 Simple Picture Graphs

May 30th, 2017 Math In Focus Ch. 11 Lesson 2More Picture Graphs

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May 31st, 2017 Math In Focus Ch. 11 Lesson 2More Picture Graphs

June 1st, 2017 Math In Focus Ch. 11 Lesson 3Tally Charts and Bar GraphsJune 2nd, 2017 Math In Focus Ch. 11Put On Your Thinking CapJune 5th, 2017 Math In Focus Ch. 11 Chapter Wrap UpJune 6th, 2017 Math In Focus Ch. 11 Test PrepMath In Focus Ch. 11 Performance TaskJune 7th, 2017 Spiral Review: Math WorkstationsJune 8th , 2017 Spiral Review: Math WorkstationsJune 9th , 2017 Spiral Review: Math WorkstationsJune 12th, 2017 DISTRICT END OF YEAR ASSESSMENTJune 13th, 2017 DISTRICT END OF YEAR ASSESSMENTJune 14th, 2017 DISTRICT END OF YEAR ASSESSMENTJune 15th, 2017 DISTRICT END OF YEAR ASSESSMENTJune 16th, 2017 DISTRICT END OF YEAR ASSESSMENTJune 19th, 2017 DISTRICT END OF YEAR ASSESSMENT END OF MP

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First Grade 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

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

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

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

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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mathematical reasoning and understanding. mathematical reasoning and understanding. progression of mathematical reasoning and understanding. Contains minor errors.progression of 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? What did you hear and see students doing in their reading and writing? 73

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%):

DEVELOPING (67% - 85%):

INSECURE (51%-65%):

BEGINNING (0%-50%):

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Student Conference Form SCHOOL: ______________________________________ TEACHER: __________________________Student Name: __________________________________________________________________________ Date: ____________________________NJSLS: ACTIVITY OBSERVED:OBSERVATION NOTES:

FEEDBACK GIVEN:

GOAL SET:

NEXT STEPS:

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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.80

MATHEMATICIAN: ____________________________________________________ SCHOOL: _______________________________ TEACHER: _________________________________ DATE: __________MATH PORTFOLIO REFLECTION FORM

PORTFOLIO ARTIFACT: _________________________________________________________________________________________________THIS IS AN EXAMPLE OF THE WORK THAT I AM MOST PROUD OF BECAUSE…..__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

_____________________________________________________________________________________________________THIS WORK ALSO SHOWS THAT I NEED TO WORK ON…______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

_____________________________________________________________________________________________________

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Resources

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Partial Sums Method

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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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