Math Unit of Study Electronic Form

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1 Grade level/course: 2 nd Grade/Math Trimester: 2 Unit of study number: 2.6 Unit of study title: Addition and Subtraction Using Strategies Number of days for this unit: 10 days (60 minutes per day) Common Core State Standards for Mathematical Content Operations and Algebraic Thinking - 2.OA Represent and solve problems involving addition and subtraction. 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. 1 NOTE: 1 See Glossary, Table 1. Number and Operations in Base Ten - 2.NBT Use place value understanding and properties of operations to add subtract. 2. NBT.9. Explain why addition and subtraction strategies work, using place value and the properties of operations. 3 NOTE: 3 Explanations may be supported by drawings or objects. Common Core State Standards for Mathematical Practice: 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to Albuquerque Public Schools June

2 help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Clarifying the Standards Prior Learning In 1 st Grade, students worked on basic one-step word problems, adding and subtracting within 20. They used mental and place value strategies to add and subtract within 100. In 1 st Grade, students also applied properties of operations to solve addition and subtraction problems (1. OA.1) (1.OA.3) (1.O.A.6). In 1 st Grade, students had exposure to addition and subtraction within 100 using concrete models, place-value, properties of operations, and/or the relationship to addition and subtraction. Students explained their reasoning by relating the strategy to a written method (1.NBT.4). Albuquerque Public Schools June

3 Current Learning This is a critical focus of instruction for 2 nd grade. This unit builds on prior knowledge developed in units 3, 4, and 5. In Unit 6, students expand their problem-solving skills into two-step word problems up to 100. Students also solve word problems with unknowns in all positions and write equations with a symbol for the unknown number to represent the problem. By Unit 15, students know all sums of two digit numbers, and can fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. By the end of this unit students can clearly explain why addition and subtraction strategies work using place-value and the properties of operations, to include the associative and commutative properties of addition. (Reference table 3 in the CCSS glossary, p. 90) When learning the properties of operations, students are not expected to name the properties used, but should be able to demonstrate the concepts. Unit 3 provides suggested developmental activities. Future Learning In 3 rd grade, students will expand their fluency into 1,000 with addition and subtraction. (The standard algorithm is introduced and secured in 4 th grade). Students will also expand arithmetic relationships into multiplication and division and apply properties of operations to multiply and divide. (3.NBT.2) (3.OA.5) (3.OA.7) Additional Findings According to Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics (2006) students solve problems by applying their understanding of the models of addition and subtraction by combining or separating sets using place-value and the commutative and associative properties. (p. 27) Further, in Principles and Standards for School Mathematics (2000) by grade 2, teachers should be encouraging students to shift towards using pencil and paper or mental models when solving mathematical problems (p. 84). Students also need to have meaningful practice to secure their ability to " develop fluency with basic number combinations and strategies with multidigit numbers." (p. 87) Content to be learned Add and subtract within 100 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 Use drawings to solve addition and subtraction problems Albuquerque Public Schools June

4 Write equations to represent problems Use place-value strategies and properties of operations to explain why addition and subtraction strategies work Mathematical practices to be integrated 1 Make sense of problems and persevere in solving them. Read the word problem and plan a solution pathway for the problem Use pictures, objects, and equations to help conceptualize and solve problems. Understand the approaches of others who may be using different methods to solve the same problem. 2 Reason abstractly and quantitatively. Decontextualize and contextualize given situations. Represent given situations involving known quantities and relationships. 4 Model with mathematics. Represent problems with drawings, objects, and equations. Apply mathematics to solve real world problems. Identify important quantities within a word problem. Essential Questions What are some different ways you can solve an addition problem? What are some different ways you can solve a subtraction problem? How can you solve an addition or subtraction problem with an unknown in different places? How can you write an equation to represent an addition or subtraction problem? Albuquerque Public Schools June

5 Grade level/course: 2 nd Grade/Math Trimester: 2 Unit of study number: 2.7 Unit of study title: Comparing Numbers to 1,000 Number of days for this unit: 10 days (60 Minutes per day) Common Core State Standards for Mathematical Content Number and Operations in Base Ten - 2.NBT Understand place value. 2. NBT.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded form. 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. Common Core State Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as Albuquerque Public Schools June

6 seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Clarifying the Standards Prior Learning In 1 st Grade, students composed numbers up to 120 using only written numerals. Students also compared two 2-digit numbers based on the meanings of tens and ones, recording their results with the symbols >, =, and <. (1.NBT.1) (1.NBT.2b) Current Learning This is a Critical Focus area, in which the students are exposed to number names and expanded notation. In Unit 7, students extend their understanding of the base-ten system by expanding notation into three places to include hundreds. Students continue to compare these numbers using the symbols >, =, and <. By the end of this unit, students can demonstrate an understanding of multi-digit numbers (up to 1,000) using base-ten numerals, number names, and expanded form. Future Learning In 3 rd Grade, students will apply their understanding of comparing numbers to explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. Students will also use their understanding of place value to round whole numbers to the nearest 10 or 100. (3.NBT.1) (3.NF.3d) Additional Findings According to A Research Companion to Principles and Standards for School Mathematics (2003), It is absolutely essential that students develop a solid understanding of the base-ten numeration system and place-value concepts by the end of grade 2. (p 81) According to Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics (2006), Children use place value and properties of operations to create equivalent representations of given numbers (such as 35 represented by 35 ones, 3 tens Albuquerque Public Schools June

7 and 5 ones, or 2 tens and 15 ones) and to write, compare, and order multidigit numbers. They use these ideas to compose and decompose multidigit numbers. (p 14) Content to be learned Read numbers to 1,000 using digits and number names Write numbers to 1,000 using digits and number names Use base-ten numerals, number names, and expanded form Compare two three-digit numbers Record results of comparisons using >, =, and < symbols Mathematical practices to be integrated 1 Make sense of problems and persevere in solving them. Show and explain to others how a problem is solved Use concrete objects or pictures to help solve a problem 7 Look for and make use of structure. Understand that a three-digit number is composed of several numbers (i.e. 609 is 6 hundreds, 0 tens, and 9 ones.) Essential Questions How do you write a three-digit number using base-ten numerals? How do you write a three-digit number using expanded form? How do you write a three digit number s name? How can you write a given number two different ways? What symbols can you use to compare numbers? Albuquerque Public Schools June

8 Grade level/course: 2nd Grade/Math Trimester: 2 Unit of study number: 2.8 Unit of study title: Addition and Subtraction Strategies Within 1000 Number of days for this unit: 10 days (60 minutes per day) Common Core State Standards for Mathematical Content Number and Operations in Base Ten 2 NBT Use place value understanding and properties of operations to add subtract. 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. 2. NBT.8 Mentally add 10 or 100 to a given number , and mentally subtract 10 or 100 from a given number Common Core State Standards for Mathematical Practice 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Albuquerque Public School - June

9 Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Clarifying the Standards Prior Learning Students in 1 st grade learned that the two-digits of a two-digit number represent amounts of tens and ones. (1.NBT.2abc) In 1st grade, students added within 100, added a two-digit number and a one digit number, and used concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. (1.NBT.4) First graders also learned to mentally find ten more or ten less than a given two digit number and explained their reasoning used. (1.NBT.5) Current Learning Students understand how to compose and decompose numbers into ones, tens, and hundreds. Students add up to four two-digit numbers using strategies based on place values and properties of operations. Students add and subtract within 1,000 using concrete models or drawings and strategies based on place value and properties of operations to do so. They also relate addition and subtraction strategies to a written method (i.e., words, pictures and/or numbers). Students explain addition and subtraction strategies using place-value, properties of operations, and/or the relationship between addition and subtraction, supported by drawings objects, or technology. Refer to Table 1 Common Addition and Subtraction Situations for additional information on the progression of the standard. (Glossary for CCSS, p. 88) By the end of this unit students are able to mentally add 10 or 100 to a given number , and can mentally subtract 10 or 100 from a given number Future Learning Fluency in adding and subtracting within 1,000 is addressed in 3rd grade (3.NBT.2) Students will also use their knowledge of place value in 3rd grade to round whole numbers to the nearest ten or hundred (3.NBT.1) Students will also assess the reasonableness of answers using mental computation and estimation strategies including rounding, using whole numbers. (3.OA.8) Albuquerque Public School - June

10 Additional Findings Effective methods for teaching multidigit addition and subtraction: According to A Research Companion to Principles and Standards for School Mathematics(2003)(p. 79), there are 3 effective methods students use to solve multidigit addition and subtraction problems, they are: 1. Counting lists methods is an extension of single digit counting where students begin counting- on by tens and ones, this allows them to be more accurate and effective. This method also naturally progresses to adding on by 100s. This is an intermediate step toward full fluency. 2. "In decomposing methods, children decompose numbers so that they can add or subtract the like units (e.g., add tens to tens, ones to ones, hundreds to hundreds, etc.)" 3. In recomposing methods, students change both numbers making it a more user friendly problem (e.g , the 22 gives 2 to change the 48 to a =70). In addition to pencil and paper tasks, as the new material is manipulated, students can be observed and questioned orally to evaluate progress with composing and decomposing. In order to emphasize problem solving, students are able to construct their own problems using the standard. The levels of complexity of addition and subtraction problems are located in Table 1 on page 7 of Progressions of the Common Core State Standards in Mathematics, K, Counting and Cardinality; K-5 Operations and Algebraic Thinking (draft) (2011)(p. 6-9). For the methods that students use for solving single digit addition and subtraction problems, refer to the grey box on page 6 of the same document. "Students can learn to compute accurately and efficiently through regular experience with meaningful procedures. They benefit from instruction that blends procedural fluency and conceptual understanding. (Principles and Standards for School Mathematics (2000), p. 87) Content to be learned Add and subtract within 1,000 using drawings, place value, and/or concrete models Use a written method to describe strategies Demonstrate the ability to compose and decompose numbers when adding and subtracting Mentally add 10 or 100 to a given number Mentally subtract 10 or 100 from a given number Mathematical practices to be integrated 2 Reason abstractly and quantitatively. Represent concrete mathematical problems with symbols Albuquerque Public School - June

11 Mentally create coherent representations of mathematical situations involving adding and/or subtraction ten or 100 Attend to the meaning of quantities when doing mental math Know and flexibly use different properties of operations 4 Model with mathematics. Write addition and subtraction equations to describe mathematical situations Analyze strategy, decide if the result makes sense, and improve strategy if needed Essential Questions What are some different ways you can solve addition and subtraction problems within 1,000? How do you use place-value, drawings, or objects to solve addition and subtraction problems within 1,000? How can you use writing to describe how you solve addition or subtraction problems? What is your strategy for adding 10 to a number? 100? What is your strategy for subtracting 10 from a number? 100? Albuquerque Public School - June

12 Grade level/course: 2nd Grade/Math Trimester: 2 Unit of study number: 2.9 Unit of study title: Introduction to Standard Units of Measurement Number of days for this unit: 10 days (60 minutes per day) Common Core State Standards for Mathematical Content Measurement and Data -- 2.MD Measure and estimate lengths in standard units 2. MD.1 Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes. 2. MD.2 Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen. 2. MD.3 Estimate lengths using units of inches, feet, centimeters, and meters. 2. MD.4 Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit. Common Core State Standards for Mathematical Practice 3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Albuquerque Public Schools - June

13 Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Clarifying the Standards Prior Learning Linear measurement with nonstandard units was a critical area of study in first grade. Students ordered objects by length (1.MD.1) and used nonstandard units to accurately measure length of an object in whole units. (1.MD.2) Current Learning Using standard units of measure is a critical area of study in second grade. Students recognize the need for standard units of measure and they select and use appropriate tools to measure length in centimeters, inches, feet, and meters). Students understand and use iteration (the act of repeating) to accurately measure the length of objects. Students estimate lengths using standard units of measure. By the end of second grade students can Albuquerque Public Schools - June

14 measure two objects of differing lengths and express the length difference in terms of a standard unit of measurement. Future Learning In third grade students will distinguish between linear and area measures. (Grade 3 overview) Third graders will measure intervals of time, liquid volume, and masses of objects. (3.MD.1, 3.MD.2) Students will measure fractions of an inch. (3.MD.4) Additional Findings Research shows that some of the most important conceptual foundations for measurement include: (1*) Unit-attribute relations, (2*) iteration, (3) tiling, (4*) identical units, (5*) standardization, (6*) proportionality, (7*) additivity, (8*) zero-point origin. (A Research Companion to Principle and Standards for School Mathematics,2003,p.180,181) NOTE: *Seven of the eight topics could relate directly to the content of this unit. "Using tools accurately and questioning when measurements may not be accurate require concepts and skills that develop over extended periods through many varied experiences." (Principles and Standards for School Mathematics,2000, p.106) Content to be learned Select and use appropriate tools to measure the length of an object. Measure objects twice using different units of measure. Estimate lengths using units of inches, feet, centimeters, and meters. Measure two objects of differing lengths and express the length difference in terms of a standard unit of measurement. Mathematical practices to be integrated 3 Construct viable arguments and critique the reasoning of others. Students are able to analyze situations involving measurement of length and use their understanding to select appropriate measurement tools. When estimating students can use concrete referents to justify their conclusions and communicate them to others. 5 Use appropriate tools strategically. Students consider all available tools when solving measurement problems and decide when a particular tool would be most helpful. When measuring, students begin to detect possible errors by strategically using estimation and other mathematical knowledge. 6 Attend to precision. Albuquerque Public Schools - June

15 Students are careful about specifying units of measure. Students measure lengths in whole standard units accurately and efficiently. Essential Questions How do I know what tool to use when measuring the length of an object? How is it possible that the same object can have two different length measurements? How could you estimate an objects' length in standard units of measure? How can you measure the difference in length of two objects in standard units? How you know you have measured an object accurately? Albuquerque Public Schools - June

16 Grade level/course: 2nd Grade/Math Trimester: 2 Unit of study number: 2.10 Unit of study title: Time is Money Number of days for this unit: 10 days (60 minutes per day) Common Core State Standards for Mathematical Content Number and Operations in Base Ten - 2.NBT Understand place value. 2. NBT.2 Count within 1000; skip-count by 5s, 10s, and 100s. Measurement and Data - 2.MD Work with time and money. 2. MD.7 Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m. 2. MD.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have? Common Core State Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Albuquerque Public Schools - June

17 Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Clarifying the Standards Prior Learning In 1 st Grade, students counted to 120. (1.NBT.1) Students also told and wrote time to hours and half hours using analog and digital clocks. (1.MD. 3) Current Learning Counting within 1,000 is a critical area for second grade learning. In this unit students gain additional practice in skip-counting by 5's, 10s, and 100 s within 1,000. The studies of time and money are not critical learning areas in second grade. In Unit 2, students tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m. Albuquerque Public Schools - June

18 Students solve word problems involving quarters, dimes, nickels, and pennies using the cents symbols ( ) appropriately. Dollars are addressed in unit 11. Future Learning In 3 rd Grade, students will tell and write time to the nearest minute and measure time to the nearest minute (3.MD.1). In 4 th Grade, students will extend problem solving to multi-step word problems, expressing money in larger quantities. (4.MD.2) Additional Findings "Students begin to work towards multiplication when they skip count by 5s, by 10s, and by 100s. This skip counting is not yet true multiplication because students don't keep track of the number of groups they have counted." (Progressions for the Common Core State Standards in Mathematics (draft), Number and Operations in Base Ten,2011, p. 8) "Time and money are wonderful and important components of any primary grade mathematics experience; however they are not points of focus in the sense of the Curriculum Focal Points concept. In fact, one could say that time and money could (maybe should) provide the context for number and algebra focal topics at these levels. As for when these are taught, certainly they are introduced early and extended through the grades, when the sophistication of time zones and monetary exchange serving as valuable contexts at the intermediate grade levels." (Curriculum Focal Points, Questions and Answers, National Council of Teachers of Mathematics, Retrieved from ( Although students may have had exposure to the use of money at home or in other curriculum areas, the subject appears for the first time in the math standards in second grade and is addressed for the first time in this unit. Students may not have an understanding of coin names or values, therefore, developmental activities should include using models that can be manipulated and studied. Teachers should concentrate on preventing misconceptions instead of correcting them and using concrete objects and pictures before introducing symbols. This will help students build strong skills in using money. Content to be learned Skip count by 5 s, 10 s and 100 s within 1,000 Tell and write time to the nearest five minutes using both analog and digital clocks Use a.m./p.m. abbreviations appropriately Solve word problems using quarters, dimes, nickels, and pennies Use symbol appropriately Albuquerque Public Schools - June

19 Mathematical practices to be integrated 1 Make sense of problems and persevere in solving them. Students look for entry points to solving time and money problems using skipcounting strategies (i.e. counting by 5s for time and nickels; counting by 10s for dimes). Students may rely on concrete objects or pictures to help conceptualize and solve a problem (i.e. coins and clocks). 6 Attend to precision. Students are careful about specifying units of measure to clarify the correspondence with quantities and problems. Students calculate accurately and efficiently expressing numerical answers with a degree of precision appropriate to problem context. 8 Look for and express regularity in repeated reasoning. Students notice if calculations are repeated, and look both for general methods and for shortcuts. Students continually evaluate the reasonableness of their intermediate results. Essential Questions What are some quick ways to count to 1,000? How can you use skip counting by 5s to tell time? How do you know what time it is? How does knowing how to skip count help you solve word problems involving money? What strategies can you use to solve word problems involving money? When telling time, what do a.m. and p.m. mean? Albuquerque Public Schools - June