6th Grade Mathematics - Orange Board of Education



2nd Grade Mathematics

Sums and Differences to 100

Understanding Place Value

Unit 2 Curriculum Map: November 14th 2016 - January 26th 2017

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Table of Contents

|I. |Mathematics Mission Statement |p. 2 |

|II. |Mathematical Teaching Practices |p. 3 |

|III. |Mathematical Goal Setting |p. 4 |

|IV. |Reasoning and Problem Solving |p. 6 |

|V. |Mathematical Representations |p. 7 |

|VI. |Mathematical Discourse |p. 9 |

|VII. |Conceptual Understanding |p. 14 |

|VIII. |Evidence of Student Thinking |p. 15 |

|IX. |Second Grade Unit II NJSLS |p. 17 |

|X. |Eight Mathematical Practices |p. 28 |

|XI. |Ideal Math Block |p. 31 |

|XII. |Math Workstations |p. 32 |

|XIII. |Math In Focus Lesson Structure |p. 35 |

|XIX. |Ideal Math Block Planning Template |p. 38 |

|XX. |Planning Calendar |p. 41 |

|XXI. |Instructional and Assessment Framework |p. 43 |

|XXII. |PLD Rubric |p. 46 |

|XXIII. |Data Driven Instruction |p. 47 |

|XXIV. |Math Portfolio Expectations |p. 50 |

Office of Mathematics Mission Statement

The 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 Curriculum

The 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 Handbook

Research 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.

Curriculum Unit Plans

Designed 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.

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.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.

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.

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:

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).

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

| | | |

|“I CAN…..” |STUDENT FRIENDLY RUBRIC |SCORE |

| | |  |

| | | | | | |

| |…a start |…getting there |…that’s it |WOW! | |

| |1 |2 |3 |4 | |

|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 |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 and say what I did and why I|  |

|or | | | |did it. | |

|Write | | |I can write or talk about why I did it. | | |

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

|Draw |thinking. |or | |thinking. | |

|or | |I can show but not draw my thinking; | | | |

|Show | | | | | |

|  | | | | | |

Reasoning and Problem Solving Mathematical Tasks

The 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.

Use and Connection of Mathematical Representations

The Lesh Translation Model

Each 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 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 Connections

As 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 Approach

The 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.

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

Mathematical Discourse and Strategic Questioning

Discourse 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?

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 Understanding

Students 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 Fluency

Procedural 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: Automaticity

Students 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

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 Strategies

First 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/or

o the relationship between addition and subtraction;

Evidence of Student Thinking

Effective 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 Proficiency

To 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.

K-2 CONCEPT MAP

|Second Grade Unit II |

|In this Unit Students will: |

| |

|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 |

| |

|2.O.A.2: |

|Fluently add and subtract within 20 using mental strategies: |

|Count On/ Count Back |

|Making Ten/Decomposing (Ten) |

|Addition and Subtraction Relationship |

|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 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. |

|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. |

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|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. |

|New Jersey Student Learning Standards: Operations and Algebraic Thinking |

|2.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) |

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

| | |

|2.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. |

| |

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|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. |

|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 ( ................
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