6th grade mathematics · web viewcreating equivalent but easier or known sums (e.g., adding 6 + 7...
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2nd Grade MathematicsUnit IV Curriculum Map: April 6th, 2017- June 19th,2017
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ORANGE PUBLIC SCHOOLSOFFICE OF CURRICULUM AND INSTRUCTIONOFFICE OF MATHEMATICS
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 and SPED Considerations p. 16X. Second Grade Unit IV NJSLS p. 24XI. Eight Mathematical Practices p. 34XII. Ideal Math Block p. 37XIII. Math In Focus Lesson Structure p. 38XIX. Ideal Math Block Planning Template p. 41XX. Planning Calendar p. 44XXI. Instructional and Assessment Framework p. 46XXII. PLD Rubric p. 49XXIII. Data Driven Instruction p. 50XXI Math Portfolio Expectations p. 53
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Office of Mathematics Mission StatementThe Office of Mathematics exists to provide the students it serves with a mathematical ‘lens’-- allowing them to better access the world with improved decisiveness, precision, and dexterity; facilities attained as students develop a broad and deep understanding of mathematical content. Achieving this goal defines our work - ensuring that students are exposed to excellence via a rigorous, standards-driven mathematics curriculum, knowledgeable and effective teachers, and policies that enhance and support learning. Office of Mathematics Objective By the year 2021, Orange Public School students will demonstrate improved academic achievement as measured by a 25% increase in the number of students scoring at or above the district’s standard for proficient (college ready (9-12); on track for college and career (K-8)) in Mathematics.Rigorous, Standards-Driven Mathematics CurriculumThe Grades K-8 mathematics curriculum was redesigned to strengthen students’ procedural skills and fluency while developing the foundational skills of mathematical reasoning and problem solving that are crucial to success in high school mathematics. Our curriculum maps are Unit Plans that are in alignment with the New Jersey Student Learning Standards for Mathematics. Office of Mathematics Department HandbookResearch tells us that teacher knowledge is one of the biggest influences on classroom atmosphere and student achievement (Fennema & Franke, 1992). This is because of the daily tasks of teachers, interpreting someone else’s work, representing and forging links between ideas in multiple forms, developing alternative explanations, and choosing usable definitions. (Ball, 2003; Ball, et al., 2005; Hill & Ball, 2009). As such, the Office of Mathematics Department Handbook and Unit Plans were intentionally developed to facilitate the daily work of our teachers; providing the tools necessary for the alignment between curriculum, instruction, and assessment. These document helps to (1) communicate the shifts (explicit and implicit) in the New Jersey Student Learning Standards for elementary and secondary mathematics (2) set course expectations for each of our courses of study and (3) encourage teaching practices that promote student achievement. These resources are accessible through the Office of Mathematics website.
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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:2.OA.1Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.2.NBT.5Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.2.NBT.9 Explain why addition and subtraction strategies work, using place value and the properties of operations.Learning Goal(s):Students will use multiple representations to solve multi-step addition and/or subtraction situations (2.OA.1) and explain the connection between various solution paths (2.NBT.5, 2.NBT.9).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.
Draw I am not able to draw or show my I can draw, but not show my I can draw and show my thinking I can draw, show and talk about 8
or Show thinking. thinking;orI can show but not draw my thinking;my thinking.
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Reasoning and Problem Solving Mathematical TasksThe benefits of using formative performance tasks in the classroom instead of multiple choice, fill in the blank, or short answer questions have to do with their abilities to capture authentic samples of students' work that make thinking and reasoning visible. Educators’ ability to differentiate between low-level and high-level demand task is essential to ensure that evidence of student thinking is aligned and targeted to learning goals. The Mathematical Task Analysis Guide serves as a tool to assist educators in selecting and implementing tasks that promote reasoning and problem solving.
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Use and Connection of Mathematical Representations
The Lesh Translation ModelEach oval in the model corresponds to one way to represent a mathematical idea. Visual: When children draw pictures, the teacher can learn more about what they understand about a particular mathematical idea and can use the different pictures that children create to provoke a discussion about mathematical ideas. Constructing their own pictures can be a powerful learning experience for children because they must consider several aspects of mathematical ideas that are often assumed when pictures are pre-drawn for students. Physical: The manipulatives representation refers to the unifix cubes, base-ten blocks, fraction circles, and the like, that a child might use to solve a problem. Because children can physically manipulate these objects, when used appropriately, they provide opportunities to compare relative sizes of objects, to identify patterns, as well as to put together representations of numbers in multiple ways.Verbal: Traditionally, teachers often used the spoken language of mathematics but rarely gave students opportunities to grapple with it. Yet, when students do have opportunities to express
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(Pictures)
(Written)
(Real Life Situations)
(Manipulatives)
(Communication)
their mathematical reasoning aloud, they may be able to make explicit some knowledge that was previously implicit for them. Symbolic: Written symbols refer to both the mathematical symbols and the written words that are associated with them. For students, written symbols tend to be more abstract than the other representations. I tend to introduce symbols after students have had opportunities to make connections among the other representations, so that the students have multiple ways to connect the symbols to mathematical ideas, thus increasing the likelihood that the symbols will be comprehensible to students. Contextual: A relevant situation can be any context that involves appropriate mathematical ideas and holds interest for children; it is often, but not necessarily, connected to a real-life situation. The Lesh Translation Model: Importance of ConnectionsAs important as the ovals are in this model, another feature of the model is even more important than the representations themselves: The arrows! The arrows are important because they represent the connections students make between the representations. When students make these connections, they may be better able to access information about a mathematical idea, because they have multiple ways to represent it and, thus, many points of access. Individuals enhance or modify their knowledge by building on what they already know, so the greater the number of representations with which students have opportunities to engage, the more likely the teacher is to tap into a student’s prior knowledge. This “tapping in” can then be used to connect students’ experiences to those representations that are more abstract in nature (such as written symbols). Not all students have the same set of prior experiences and knowledge. Teachers can introduce multiple representations in a meaningful way so that students’ opportunities to grapple with mathematical ideas are greater than if their teachers used only one or two representations.Concrete Pictorial Abstract (CPA) Instructional ApproachThe CPA approach suggests that there are three steps necessary for pupils to develop understanding of a mathematical concept. Concrete: “Doing Stage”: Physical manipulation of objects to solve math problems.
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Pictorial: “Seeing Stage”: Use of imaged to represent objects when solving math problems.Abstract: “Symbolic Stage”: Use of only numbers and symbols to solve math problems.CPA is a gradual systematic approach. Each stage builds on to the previous stage. Reinforcement of concepts are achieved by going back and forth between these representations
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Mathematical Discourse and Strategic QuestioningDiscourse involves asking strategic questions that elicit from students both how a problem was solved and why a particular method was chosen. Students learn to critique their own and others' ideas and seek out efficient mathematical solutions.While classroom discussions are nothing new, the theory behind classroom discourse stems from constructivist views of learning where knowledge is created internally through interaction with the environment. It also fits in with socio-cultural views on learning where students working together are able to reach new understandings that could not be achieved if they were working alone.Underlying the use of discourse in the mathematics classroom is the idea that mathematics is primarily about reasoning not memorization. Mathematics is not about remembering and applying a set of procedures but about developing understanding and explaining the processes used to arrive at solutions.Asking better questions can open new doors for students, promoting mathematical thinking and classroom discourse. Can the questions you're asking in the mathematics classroom be answered with a simple “yes” or “no,” or do they invite students to deepen their understanding?
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To help you encourage deeper discussions, here are 100 questions to incorporate into your instruction by Dr. Gladis Kersaint, mathematics expert and advisor for Ready Mathematics.
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Conceptual UnderstandingStudents demonstrate conceptual understanding in mathematics when they provide evidence that they can: recognize, label, and generate examples of concepts; use and interrelate models, diagrams, manipulatives, and varied representations of concepts; identify and apply principles; know and apply facts and definitions; compare, contrast, and integrate related concepts and principles; and recognize, interpret, and apply the signs, symbols, and terms used to represent concepts.
Conceptual understanding reflects a student's ability to reason in settings involving the careful application of concept definitions, relations, or representations of either.Procedural FluencyProcedural fluency is the ability to: apply procedures accurately, efficiently, and flexibly;
to transfer procedures to different problems and contexts; to build or modify procedures from other procedures; and to recognize when one strategy or procedure is more appropriate to apply than another.
Procedural fluency is more than memorizing facts or procedures, and it is more than understanding and being able to use one procedure for a given situation. Procedural fluency builds on a foundation of conceptual understanding, strategic reasoning, and problem solving (NGA Center & CCSSO, 2010; NCTM, 2000, 2014). Research suggests that once students have memorized and practiced procedures that they do not understand, they have less motivation to understand their meaning or the reasoning behind them (Hiebert, 1999). Therefore, the development of students’ conceptual understanding of procedures should precede and coincide with instruction on procedures. Math Fact Fluency: AutomaticityStudents who possess math fact fluency can recall math facts with automaticity. Automaticity is the ability to do things without occupying the mind with the low-level details required, allowing it to become an automatic response pattern or habit. It is usually the result of learning, repetition, and practice.K-2 Math Fact Fluency Expectation
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K.OA.5 Add and Subtract within 5.1.OA.6 Add and Subtract within 10.2.OA.2 Add and Subtract within 20.Math Fact Fluency: Fluent Use of Mathematical StrategiesFirst and second grade students are expected to solve addition and subtraction facts using a variety of strategies fluently.1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as:
counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction; and creating equivalent but easier or known sums.
2.NBT.7 Add and subtract within 1000, using concrete models or drawings and strategies based on:o place value, o properties of operations, and/oro the relationship between addition and subtraction;
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Evidence of Student ThinkingEffective classroom instruction and more importantly, improving student performance, can be accomplished when educators know how to elicit evidence of students’ understanding on a daily basis. Informal and formal methods of collecting evidence of student understanding enable educators to make positive instructional changes. An educators’ ability to understand the processes that students use helps them to adapt instruction allowing for student exposure to a multitude of instructional approaches, resulting in higher achievement. By highlighting student thinking and misconceptions, and eliciting information from more students, all teachers can collect more representative evidence and can therefore better plan instruction based on the current understanding of the entire class. Mathematical ProficiencyTo be mathematically proficient, a student must have: • Conceptual understanding: comprehension of mathematical concepts, operations, and relations; • Procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently, and appropriately; • Strategic competence: ability to formulate, represent, and solve mathematical problems; • Adaptive reasoning: capacity for logical thought, reflection, explanation, and justification; • Productive disposition: habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy.Evidence should:
Provide a window in student thinking; Help teachers to determine the extent to which students are reaching the math learning goals; and Be used to make instructional decisions during the lesson and to prepare for subsequent lessons.
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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.23
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.
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 pairs
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Steps 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.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.
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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.
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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: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 another28
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Scaffolding: 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 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, 201630
K-2 CONCEPT MAP
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Second Grade Unit IVIn this Unit Students will:2.OA.1:
Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of:o Adding to,o Taking from,o Putting Together,o Taking Apart, ando Comparing with unknowns in all positions
2.O.A.2: Fluently add and subtract within 20 using mental strategies:
o Count On/ Count Backo Making Ten/Decomposing (Ten)o Addition and Subtraction Relationshipo Doubles +/-
Know from memory all sums of two one digit numbers. 2.NBT.1-9
Extend their concept of numbers Gain knowledge of how to count, read, and write up to 1,000 Use Base-ten blocks, place-value charts, and number lines to develop the association between
the physical representation of the number, the number symbol, and the number word Compose and decompose numbers through place value, number bonds Apply place value in addition with and without regrouping in numbers up to 1000 Use multiple strategies: concrete, pictorial and abstract representations.
Mathematical Practices Make sense of problem solving 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
2.OA.1
Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.Second Grade students extend their work with addition and subtraction word problems in two major ways. First, they represent and solve word problems within 100, building upon their previous work to 20. In addition, they represent and solve one and two-step word problems of all three types (Result Unknown, Change Unknown, Start Unknown). Please see Table 1 at end of document for examples of all problem types.One-step word problems use one operation. Two-step word problems use two operations which may include the same operation or opposite operations.
Two-Step Problems: Because Second Graders are still developing proficiency with the most difficult subtypes (shaded in white in Table 1 at end of the glossary): Add To/Start Unknown; Take From/Start Unknown; Compare/Bigger Unknown; and Compare/Smaller Unknown, two-step problems do not involve these sub-types (Common Core Standards Writing Team, May 2011). Furthermore, most two-step problems should focus on single-digit addends since the primary focus of the standard is the problem-type.33
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New Jersey Student Learning Standards: Operations and Algebraic Thinking2.OA.2
Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers. See standard 1.OA.6 for a list of mental strategies.Building upon their work in First Grade, Second Graders use various addition and subtraction strategies in order to fluently add and subtract within 20:1.OA.6 Mental Strategies
Counting On/Counting Back 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) Creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12, 12 + 1 = 13
Second Graders internalize facts and develop fluency by repeatedly using strategies that make sense to them. When students are able to demonstrate fluency they are accurate, efficient, and flexible. Students must have efficient strategies in order to know sums from memory.Research indicates that teachers can best support students’ memory of the sums of two one-digit numbers through varied experiences including making 10, breaking numbers apart, and working on mental strategies. These strategies replace the use of repetitive timed tests in which students try to memorize operations as if there were not any relationships among the various facts. When teachers teach facts for automaticity, rather than memorization, they encourage students to think about the relationships among the facts. (Fosnot & Dolk, 2001)It is no accident that the standard says “know from memory” rather than “memorize”. The first describes an outcome, whereas the second might be seen as describing a method of achieving that outcome. So no, the standards are not dictating timed tests. (McCallum, October 2011)
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New Jersey Student Learning Standards: Numbers and Operations in Base Ten2.NBT.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones.
Second Grade students extend their base-ten understanding to hundreds as they view 10 tens as a unit called a “hundred”. They use manipulative materials and pictorial representations to help make a connection between the written three-digit numbers and hundreds, tens, and ones.
Second Graders extend their work from first grade by applying the understanding that “100” is the same amount as 10 groups of ten as well as 100 ones. This lays the groundwork for the structure of the base-ten system in future grades. Second Grade students build on the work of 2.NBT.2a. They explore the idea that numbers such as 100, 200, 300, etc., are groups of hundreds with zero tens and ones. Students can represent this with both groupable (cubes, links) and pregrouped (place value blocks) materials.
2.NBT.2 Count within 1000; skip-count by 5s, 10s, and 100s.
Second Grade students count within 1,000. Thus, students “count on” from any number and say the next few numbers that come afterwards.Example: What are the next 3 numbers after 498? 499, 500, 501.When you count back from 201, what are the first 3 numbers that you say? 200, 199, 198.Second grade students also begin to work towards multiplication concepts as they skip count by 5s, by 10s, and by 100s. Although skip counting is not yet true multiplication because students don’t keep track of the number of groups they have counted, they can explain that when they count by 2s, 5s, and 10s they are counting groups of items with that amount in each group.
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As teachers build on students’ work with skip counting by 10s in Kindergarten, they explore and discuss with students the patterns of numbers when they skip count. For example, while using a 100s board or number line, students learn that the ones digit alternates between 5 and 0 when skip counting by 5s. When students skip count by 100s, they learn that the hundreds digit is the only digit that changes and that it increases by one number. 2.NBT.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
Second graders read, write and represent a number of objects with a written numeral (number form or standard form). These representations can include snap cubes, place value (base 10) blocks, pictorial representations or other concrete materials. Please be cognizant that when reading and writing whole numbers, the word “and” should not be used (e.g., 235 is stated and written as “two hundred thirty-five).Expanded form (125 can be written as 100 + 20 + 5) is a valuable skill when students use place value strategies to add and subtract large numbers in 2.NBT.7.
2.NBT.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.
Second Grade students build on the work of 2.NBT.1 and 2.NBT.3 by examining the amount of hundreds, tens and ones in each number. When comparing numbers, students draw on the understanding that 1 hundred (the smallest three-digit number) is actually greater than any amount of tens and ones represented by a two-digit number. When students truly understand this concept, it makes sense that one would compare three-digit numbers by looking at the hundreds place first.Students should have ample experiences communicating their comparisons in words before using symbols. Students were introduced to the symbols greater than (>), less than (<) and equal to (=) in First Grade and continue to use them in Second Grade with numbers within 1,000.Example: Compare these two numbers. 452 __ 455
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While students may have the skills to order more than 2 numbers, this Standard focuses on comparing two numbers and using reasoning about place value to support the use of the various symbols.2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
There are various strategies that Second Grade students understand and use when adding and subtracting within 100 (such as those listed in the standard). The standard algorithm of carrying or borrowing is neither an expectation nor a focus in Second Grade. Students use multiple 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: 67 + 25 = __
Example: 63 – 32 = __39
2.NBT.6 Add up to four two-digit numbers using strategies based on place value and properties of operations.
Second Grade students add a string of two-digit numbers (up to four numbers) by applying place value strategies and properties of operations. Example: 43 + 34 + 57 + 24 = __
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2.NBT.7 Add and subtract within 1000, 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. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
Second graders extend the work from 2.NBT. to two 3-digit numbers. Students should have ample experiences using concrete materials and pictorial representations to support their work. This standard also references composing and decomposing a ten. This work should include strategies such as making a 10, making a 100, breaking apart a 10, or creating an easier problem.
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The standard algorithm of carrying or borrowing is not an expectation in Second Grade. Students are not expected to add and subtract whole numbers using a standard algorithm until the end of Fourth Grade. Example: 354 + 287 = __
2.NBT.8 Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100-900.
Second Grade students mentally add or subtract either 10 or 100 to any number between 100 and 900. As teachers provide ample experiences for students to work with pre-grouped objects and facilitate discussion, second graders realize that when one adds or subtracts 10 or 100 that only the tens place or the digit in the hundreds place changes by 1. As the teacher facilitates opportunities for patterns to emerge and be discussed, students notice the patterns and connect the digit change with the amount changed. 42
Opportunities to solve problems in which students cross hundreds are also provided once students have become comfortable adding and subtracting within the same hundred.Example: Within the same hundred What is 10 more than 218? What is 241 – 10? Example: Across hundreds 293 + 10 = ☐ What is 10 less than 206?This standard focuses only on adding and subtracting 10 or 100. Multiples of 10 or multiples of 100 can be explored; however, the focus of this standard is to ensure that students are proficient with adding and subtracting 10 and 100 mentally.
2.NBT.9 Explain why addition and subtraction strategies work, using place value and the properties of operations.
Second graders explain why addition or subtraction strategies work as they apply their knowledge of place value and the properties of operations in their explanation. They may use drawings or objects to support their explanation. Once students have had an opportunity to solve a problem, the teacher provides time for students to discuss their strategies and why they did or didn’t work.
Example: There are 36 birds in the park. 25 more birds arrive. How many birds are there? Solve the problem and show your work.
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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 quantitatively45
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 mathematics such as determining the length of quantities when measuring with standard units.
3
Construct 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 46
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 strategically
Mathematically 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 to use than a ruler.
6Attend to precision
Mathematically 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.
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7
Look for and make use of structure
Mathematically 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?”
8 Look 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
GUIDED LEARNING: Partner Math 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:
MobyMax
Math Journal
LabFluency Lab:First In Math
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 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, Grade 2 – 5 Transitional Guide is located on ThinkCentral.com 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:2.OA.2
Strategy:
Tool(s):
Mat
h In
Foc
us/E
nGag
eNY
Launch
Exploration
Independent Practice
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Diffe
rent
iatio
n: M
ath
Wor
ksta
tions
Small Group Instruction
Tech. Lab
Problem Solving Lab
CCSS:2.OA.12.NBT.62.NBT.7
Fluency Lab2.OA.22.NBT.52.NBT.8
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 PreparationLesson Planning Support Tool______________________________________________________________________________________________________Component 1A: Knowledge of Content and PedagogyContent
Fluency Practice and Anchor Problem clearly outlined in lesson plans provide reinforcement of prerequisite knowledge/skills needed;Essentials question(s) and lesson objective(s) support learning of New Jersey Student Learning Standards grade level expectations;PedagogyDaily fluency practice is clearly outlined in lesson plans;Multiple strategies are evident within lesson plans;Mathematical tools outlined within lesson plans;___________________________________________________________________________________________________________________________Component 1B: Knowledge of StudentsIntentional Student Grouping is evident within lesson plans: Independent Practice: Which students will work on:
MIF Re-TeachMIF PracticeMIF Extra PracticeMIF EnrichmentMath Workstations: Which students will work in:Fluency LabTechnology LabMath JournalProblem Solving Lab
Component 1C: Setting Instructional OutcomesLesson plan objectives are aligned to one or more New Jersey Student Standards for Learning;Connections made to previous learning;Outcomes: student artifacts are differentiated;
Component 1D: Demonstrating Knowledge of ResourcesDistrict Approved Programs: Use Math In Focus/EnGageNY/Go Math resources are evident;Technology: Technology used to help students understand the lesson objective is evident; Students use technology to gain an understanding of the lesson objective;Supplemental Resources: Integration of additional materials evident (Math Workstations)________________________________________________________________________________________________________________________Component 1E: Designing Coherent InstructionLesson Plans support CONCEPTUAL UNDERSTANDING;
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Lesson Plans show evidences of CONCRETE, PICTORIAL, and ABSTRACT representation;Alignment between OBJECTIVES, APPLICATION, and ASSESSMENT evident;___________________________________________________________________________________________________________________________Component 1F: Assessing Student LearningLesson Plans include: Focus Question/Essential Understanding Anchor Problem Checks for Understanding Demonstration of Learning (Exit Ticket)
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Planning Calendar April 2017
Monday Tuesday Wednesday Thursday Friday3 4 5 6 7
ENGAGENY Module 4 End of Module Assessment10
ENGAGENY Module 4 End of Module Assessment
11 12 13 14
17 18 19 20 21
24 25 26 27 28
May 2017Monday Tuesday Wednesday Thursday Friday
1 2 3 4 58 9 10 11 1215 16 17 18 1922 23 24 25 26
29 30 31
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Planning Calendar June 2017Monday Tuesday Wednesday Thursday Friday
1 25 6 7 8 912 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
Second Grade Unit IV Instructional and Assessment FrameworkRecommended Activities CCSS Notes April 10th –June 19, 2016
Utilize DAILY FLUENCY (5 minutes) and FLUENCY WORKSTATION (25 minutes) to address grade level TIME expectations.Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.
2.MD.7 Project Based Learning Opportunity: TimeDevelop a project that allows students to make real world connections.Provide grading expectations in advance (rubric).April 10th- , 2017 Math In Focus Ch. 17 OpenerRecall Prior KnowledgeCh. 17 Pre-Test
2.MD.92.MD.10
Project Based Learning Opportunity: DataDevelop a project that allows students to make real world connections.Provide grading expectations in advance (rubric).April 11th- , 2017 Math In Focus Ch. 17 Lesson 1
Reading Picture Graphs with Scales
April 12th- , 2017 Math In Focus Ch. 17 Lesson 2Making Picture Graphs
April 13th- , 2017 Math In Focus Ch. 17 Lesson 3Real World Problems: Picture GraphsApril 14th – 21st, 2017 Spring RecessApril 24th , 2017 Math In Focus Ch. 17 Put On Your Thinking CapApril 25th , 2017 Math In Focus Ch. 17 Chapter Wrap UpApril 26th , 2017 Math In Focus Ch. 17 Test PrepMath In Focus Ch. 17 Performance TaskApril 27th , 2017 EnGageNY Module 5: Topic A: Lesson 1Relate 10 more, 10 less, 100 more, and 100 less to addition and subtraction of 10 and 100.April 28th , 2017 EnGageNY Module 5: Topic A: Lesson 2Add and subtract multiples of 100, including counting on to subtract.
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2.NBT.7 2.NBT.8 2.NBT.9May 1st , 2017 EnGageNY Module 5: Topic A: Lesson 3Add multiples of 100 and some tens within 1,000.May 2nd, 2017 EnGageNY Module 5: Topic A: Lesson 4Subtract multiples of 100 and some tens within 1,000.May 3rd, 2017 EnGageNY Module 5: Topic A: Lesson 5Use the associative property to make a hundred in one addend.May 4th , 2017 EnGageNY Module 5: Topic A: Lesson 6Use the associative property to subtract from three-digit numbers and verify solutions with addition.May 5th , 2017 EnGageNY Module 5: Topic A: Lesson 7Share and critique solution strategies for varied addition and subtraction problems within 1,000.May 8th, 2017
EnGageNY Module 5: Topic B: Lesson 8Relate manipulative representations to the addition algorithm2.NBT.7 2.NBT. 9
May 9th, 2017 EnGageNY Module 5: Topic B: Lesson 9Relate manipulative representations to the addition algorithmMay 10th, 2017 EnGageNY Module 5: Topic B: Lesson 10Use math drawings to represent additions with up to two compositions and relate drawings to the addition algorithmMay 11th, 2017 EnGageNY Module 5: Topic B: Lesson 11Use math drawings to represent additions with up to two compositions and relate drawings
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to the addition algorithmMay 12th, 2017 EnGageNY Module 5: Topic B: Lesson 12Choose and explain solution strategies and record with a written addition method.May 15th, 2017 EnGageNY Module 5: Mid-Module AssessmentMath WorkstationsMay 16th, 2017 EnGageNY Module 5: Mid-Module AssessmentMath WorkstationsMay 17th, 2017 Math In Focus Ch. 11 OpenerRecall Prior KnowledgeCh. 11 Pre-Test
2MD.8 Project Based Learning Opportunity: MoneyDevelop a project that allows students to make real world connections.Provide grading expectations in advance (rubric).
May 18th, 2017 Math In Focus Ch. 11 Lesson 1 Coins and BillsMay 19th, 2017 Math In Focus Ch. 11 Lesson 1 Coins and BillsMay 22nd, 2017 Math In Focus Ch. 11 Lesson 1 Coins and BillsMay 23rd, 2017 Math In Focus Ch. 11 Lesson 2 Comparing Amounts of MoneyMay 24th, 2017 Math In Focus Ch. 11 Lesson 3Real World Problems: MoneyMay 25th, 2017 Math In Focus Ch. 11 Put On Your Thinking Cap 2.NBT.7 2.NBT.8 2.NBT.9May 26th, 2017 Math In Focus Ch. 11 Chapter Wrap UpMay 29th, 2017 Math In Focus Ch. 11 Test PrepMath In Focus Ch. 11 Performance TaskMay 30th, 2017 Math In Focus Ch. 19 OpenerRecall Prior KnowledgeCh. 19 Pre-Test
Project Based Learning Opportunity: ShapesDevelop a project that allows students to make real world connections.Provide grading expectations in advance
May 31st, 2017 Math In Focus Ch. 19 Lesson 1Plane Shape 2.G.1June 1st, 2017 Math In Focus Ch. 19 Lesson 1Plane Shape
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(rubric).
June 2nd, 2017 Math In Focus Ch. 19 Lesson 2Solid ShapeJune 5th, 2017 Math In Focus Ch. 19 Lesson 3Making PatternsJune 6th, 2017 Math In Focus Ch. 19 Put On Your Thinking CapJune 7th, 2017 Math In Focus Ch. 19Chapter Wrap UpJune 8th , 2017 Math In Focus Ch. 19 Test PrepMath In Focus Ch. 19 Performance TaskJune 9th , 2017 Spiral Review: Math WorkshopsJune 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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Got ItEvidence shows that the student essentially has the target concept or big math idea.Not There YetStudent shows evidence of a major misunderstanding, incorrect concepts or procedure, or a failure to engage in the task.PLD Level 5: 100%Distinguished command PLD Level 4: 89%Strong Command PLD Level 3: 79%Moderate Command PLD Level 2: 69%Partial Command PLD Level 1: 59%Little CommandStudent work shows distinguished levels of understanding of the mathematics.
Student constructs and communicates a complete response based on explanations/reasoning using the: Tools:
o Manipulativeso Five Frameo Ten Frameo Number Lineo Part-Part-Whole Model
Strategies:o Drawingso Counting Allo Count On/Backo Skip Countingo Making Teno Decomposing Number
Precise use of math vocabularyResponse includes an efficient and logical progression of mathematical reasoning and understanding.
Student work shows strong levels of understanding of the mathematics. Student constructs and communicates a complete response based on explanations/reasoning using the: Tools:
o Manipulativeso Five Frameo Ten Frameo Number Lineo Part-Part-Whole Model
Strategies:o Drawingso Counting Allo Count On/Backo Skip Countingo Making Teno Decomposing Number
Precise use of math vocabularyResponse includes a logical progression of mathematical reasoning and understanding.
Student work shows moderate levels of understanding of the mathematics. Student constructs and communicates a complete response based on explanations/reasoning using the: Tools:
o Manipulativeso Five Frameo Ten Frameo Number Lineo Part-Part-Whole Model
Strategies:o Drawingso Counting Allo Count On/Backo Skip Countingo Making Teno Decomposing Number
Precise use of math vocabularyResponse includes a logical but incomplete progression of mathematical reasoning and understanding. Contains minor errors.
Student work shows partial understanding of the mathematics. Student constructs and communicates an incomplete response based on student’s attempts of explanations/ reasoning using the: Tools:
o Manipulativeso Five Frameo Ten Frameo Number Lineo Part-Part-Whole Model
Strategies:o Drawingso Counting Allo Count On/Backo Skip Countingo Making Teno Decomposing Number
Precise use of math vocabularyResponse includes an incomplete or illogical progression of mathematical reasoning and understanding.
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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5 points 4 points 3 points 2 points 1 pointSecond Grade PLD Rubric
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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?
Now it is time to begin the analysis again.65
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.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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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. 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.70
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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ADDITION FACTS WITHIN 20 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
0+2020+00+1919+0
0+1818+00+1717+0
0+1616+00+1515+0
0+1414+00+1313+0
0+1212+00+1111+0
0+1010 +0 0+99+0 0+88+0 0+77+0 0+66+0 0+55+0 0+44+0 0+33+0 0+22+0 0+11+0 0+01+1919+1
1+1818+11+1717+1
1+1616+11+1515+1
1+1414+11+1313+1
1+1212+11+1111+1
1+1010+1 1+99+1 1+88+1 1+77+1 1+66+1 1+55+1 1+44+1 1+33+1 1+22+1 1+12+1818+2
2+1717+22+1616+2
2+1515+22+1414+2
2+1313+22+1212+2
2+1111+22+1010+2 2+99+2 2+88+2 2+77+2 2+66+2 2+55+2 2+44+2 2+33+2 2+2
3+1717+33+1616+3
3+1515+33+1414+3
3+1313+33+1212+3
3+1111+33+1010+3 3+99+3 3+88+3 3+77+3 3+66+3 3+55+3 3+44+3 3+3
4+1616+44+1515+4
4+1414+44+1313+4
4+1212+44+1111+4
4+1010+4 4+99+4 4+88+4 4+77+4 4+66+4 4+55+4 4+45+1515+5
5+1414+55+1313+5
5+1212+55+1111+5
5+1010+5 5+99+5 5+88+5 5+77+5 5+66+5 5+56+1414+6
6+1313+66+1212+6
6+1111+66+1010+6 6+99+6 6+88+6 6+77+6 6+6
7+1313+77+1212+7
7+1111+77+1010+7 7+99+7 7+88+7 7+7
8+1212+8 8+11 8+10 8+99+8 8+873
11+8 10+89+1111+9
9+1010+9 9+910+10
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SUBTRACTION FACTS WITHIN 20
20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 020-0 20-1 20-2 20-3 20-4 20-5 20-6 20-7 20-8 20-9 20-10 20-11 20-12 20-13 20-14 20-15 20-16 20-17 20-18 20-19 20-20
19-0 19-1 19-2 19-3 19-4 19-5 19-6 19-7 19-8 19-9 19-10 19-11 19-12 19-13 19-14 19-15 19-16 19-17 19-18 19-1918-0 18-1 18-2 18-3 18-4 18-5 18-6 18-7 18-8 18-9 18-10 18-11 18-12 18-13 18-14 18-15 18-1 6 18-17 18-18
17-0 17-1 17-2 17-3 17-4 17-5 17-6 17-7 17-8 17-9 17-10 17-11 17-12 17-13 17-4 17-15 17-16 17-1716-0 16-1 16-2 16-3 16-4 16-5 16-6 16-7 16-8 16-9 16-10 16-11 15-12 15-13 16-14 16-15 16-16
15-0 15-1 15-2 15-3 15-4 15-5 15-6 15-7 15-8 15-9 15-10 15-11 15-12 15-13 15-14 15-1514-0 14-1 14-2 14-3 14-4 14-5 14-6 14-7 14-8 14-9 14-10 14-11 14-12 14-13 14-14
13-0 13-1 13-2 13-3 13-4 13-5 13-6 13-7 13-8 13-9 13-10 13-11 13-12 13-1312-0 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10 12-11 12-12
11-0 11-1 11-2 11-3 11-4 11-5 11-6 11-7 11-8 11-9 11-10 11-1110-0 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10
9-0 9-1 9-2 9-3 9-4 9-5 9-6 9-7 9-8 9-98-0 8-1 8-2 8-3 8-4 8-5 8-6 8-7 8-8
7-0 7-1 7-2 7-3 7-4 7-5 7-6 7-76-0 6-1 6-2 6-3 6-4 6-5 6-6
5-0 5-1 5-2 5-3 5-4 5-54-0 4-1 4-2 4-3 4-4
75
3-0 3-1 3-2 3-32-0 2-1 2-2
1-0 1-10-0
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Partial Sums Method
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OPEN NUMBER LINE What exactly is an open number line? Well, it’s a number line with no numbers or tick marks. Open number lines are great models for working with place value or, in the case below, addition. The number line is a fantastic way to record the different strategies used by students. The three number lines below all show strategies for adding 37 + 48.
Number Line 1: This student added the tens (30 + 40) and then the ones (7 + 8). The number line starts at 30 (the tens from the first number) and adds on the 4 tens from the second number, landing on 70. The student then added 7 + 8 to get 15 and added that to the 70 to get 85.Number Line 2: This student left 37 whole and added on the 4 tens from the second number. He then broke the 8 ones into 3 + 5 and used the 3 ones to make 80. Finally, he added on the remaining 5 ones.Number Line 3: This student took 3 of the 8 ones from the second number to get make a ten out of the 37 (37 + 3 = 40). Then, she jumped on the 4 tens to get to 80. Last, she added the remaining 5 ones.Notice the number sense required for this type of math. Students have to be able to think flexibly about numbers, understand place value, and decompose numbers. This might be out of your comfort zone! If so, try some problems on your own. When you do this with your class, it is a good idea to anticipate the strategies students might use, so you’ll be ready to draw them.
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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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