Gorenflo Elementary School

2nd Grade Math Pacing Guide

When a cluster is referred to in this chart without (notes), the cluster is taught in its entirety.

Term 1: Module 1, Module 2 & Start Module 3( tested term 2)

Module 1: Sums and Differences to 20

Represent and solve problems involving addition and subtraction. (In this module, word problems focus primarily on result unknown and change unknown situations.)

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. (See Glossary, Table 1.)

Add and subtract within 20. (From this point forward, fluency practice with addition and subtraction to 20 is part of the students’ ongoing experience.)

2.OA.2 Fluently add and subtract within 20 using mental strategies. (See standard 1.OA.6 for a list of mental strategies.) By end of grade 2, know from memory all sums of two one-digit numbers.

Use place value understanding and properties of operations to add and subtract. (The balance of this cluster is addressed in Modules 4 and 5.)

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.

Module 2: Addition and Subtraction of Length Units

Measure and estimate lengths in standard units. (Focus is on metric measurement in preparation for place value in Module 3. Customary measurement is addressed in Module 7.)

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.

Term 1

Relate addition and subtraction to length.

2.MD.5 Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

2.MD.6 Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

You will start Module 3 in the 1^st term, but it will be tested 2^nd Term.

Module 3: Place Value, counting, and Comparison of Numbers to 1,000

Understand place value.

2.NBT.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

a. 100 can be thought of as a bundle of ten tens – called a “hundred.”

b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

2.NBT.2 Count within 1000; skip-count by 5s(Use analog clock to provide a context for skip-counting by 5s.) 10s, and 100s.

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.

Term 1 Module 1: Focus Standards for Mathematical Practice

MP.2 Reason abstractly and quantitatively. Students reason quantitatively when they measure and compare lengths. They reason abstractly when they use estimation strategies such as benchmarks and mental rulers, and when they relate number line diagrams to measurement models.

MP.3 Construct viable arguments and critique the reasoning of others. Students reason to solve word problems involving length measurement using tape diagrams and also analyze the reasonableness of the work of their peers.

MP.5 Use appropriate tools strategically. Students consider the object being measured and choose the appropriate measurement tool. They use the tool of the tape diagram to solve word problems.

MP.6 Attend to precision. Students accurately measure by laying physical units end-to-end with no gaps and when using a measurement tool. They correctly align the zero-point on a ruler as the beginning of the total length. They attend to precision when they verbally and in writing specify the length unit, when they use a ruler to measure or draw a straight line of a given length, and when they verify estimations by measuring.

Term 1 Module 2: Focus Standards for Mathematical Practice

MP.2 Reason abstractly and quantitatively. Students reason quantitatively when they measure and compare lengths. They reason abstractly when they use estimation strategies such as benchmarks and mental rulers, and when they relate number line diagrams to measurement models.

MP.3 Construct viable arguments and critique the reasoning of others. Students reason to solve word problems involving length measurement using tape diagrams and also analyze the reasonableness of the work of their peers.

MP.5 Use appropriate tools strategically. Students consider the object being measured and choose the appropriate measurement tool. They use the tool of the tape diagram to solve word problems.

MP.6 Attend to precision. Students accurately measure by laying physical units end-to-end with no gaps and when using a measurement tool. They correctly align the zero-point on a ruler as the beginning of the total length. They attend to precision when they verbally and in writing specify the length unit, when they use a ruler to measure or draw a straight line of a given length, and when they verify estimations by measuring.

Term 2: Module 3 & Module 4

You will start Module 3 in 1^st term, but it will be tested 2^nd Term .

Module 3: Place Value, counting, and Comparison of Numbers to 1,000

Understand place value.

2.NBT.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

a. 100 can be thought of as a bundle of ten tens – called a “hundred.”

b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

2.NBT.2 Count within 1000; skip-count by 5s(Use analog clock to provide a context for skip-counting by 5s.) 10s, and 100s.

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.

Module 4: Addition and Subtraction Within 200 with Word Problems to 100

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. (See Glossary, Table 1.)

Use place value understanding and properties of operations to add and subtract. (In this module, work is limited to within 200. This work is extended to numbers within 1000 in the next module.)

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.

Term 2

2.NBT.6 Add up to four two-digit numbers using strategies based on place value and properties of operations.

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 100-900, and mentally subtract 10 or 100 from a given number 100-900.

2.NBT.9 Explain why addition and subtraction strategies work, using place value and the properties of operations. (Explanations may be supported by drawings or objects.

Term 2 Module 3: Focus Standards for Mathematical Practice

MP.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 6 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 (exemplified in Topic D).

MP.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. 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 argument (exemplified in Topics A and E).

MP.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 and 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 (exemplified in Topics C and F).

MP.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 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 7 × 5 + 7 × 3, in preparation for learning about the

distributive property. In the expression x² + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. 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 (exemplified in Topic B).

MP.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) (x² + x + 1), and (x – 1) (x³ + x² + 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 (exemplified in Topic G).

Term 2 Module 4: Focus Standards for Mathematical Practice

MP.1 Make sense of problems and persevere in solving them. Students solve two-step word problems, and are challenged to make sense of more complex relationships within situations. They flexibly solve problems with a variety of strategies at their disposal, sometimes finding many ways to solve the same problem.

MP.2 Reason abstractly and quantitatively. Students reason abstractly when they represent two-step problems and harder problem types with drawings such as tape diagrams and when they relate those drawings to equations. As the Module progresses, students move back and forth between concrete, pictorial, and abstract work to make sense of quantities and their relationships in problem situations.

MP.3 Construct viable arguments and critique the reasoning of others. Students construct viable arguments when they use place value reasoning and properties of operations to explain why their addition and subtraction strategies work, and when they use that reasoning to justify their choice of strategies in solving problems. They critique the reasoning of others when they use those same concepts to disprove or support the work of their peers.

MP.4 Model with mathematics. Students model with mathematics when they write equations to solve two-step word problems, make math drawings when solving a vertical algorithm, or when they draw place value charts and disks to represent numbers.

MP.6 Attend to precision. Students attend to precision when they label their math drawings and models with specific place value units. They calculate accurately and efficiently when adding numbers within 200 and they use the relationship between addition and subtraction to check their work.

Term 3: Module 5 & Module 6

Module 5: Addition and Subtraction Within 1000 with Word Problems to 100

Use place value understanding and properties of operations to add and subtract. (The balance of this cluster is addressed in Module 4.)

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 100-900, and mentally subtract 10 or 100 from a given number 100-900.

2.NBT.9 Explain why addition and subtraction strategies work, using place value and the properties of operations. (Explanations may be supported by drawings or objects.)

Module 6: Foundations of Multiplication and Division

Work with equal groups of objects to gain foundations for multiplication.

2.OA.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g.,

by pairing objects or counting them by 2s: write an equation to express an even number as a sum of

two equal addends.

2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows

and up to 5 columns; write an equation to express the total as a sum of equal addends.

Reason with shapes and their attributes.( 2.G.2 is taught before G.1 and G.3 because the array model is so important to

the foundation for multiplication.)

2.G.2 Partition a rectangle into rows and columns of same size squares and count to find the

total number of them.

Term 3 Module 5: Focus Standards for Mathematical Practice

MP.3 Construct viable arguments and critique the reasoning of others. Students use place value reasoning to explain how each step in their drawing relates to a step in the written method. They choose and explain various solution strategies such as number bonds, chip models, the vertical method, arrow notation, and tape diagrams. They critique the reasoning of others when they listen to their peers explain their strategies for solving problems, and they discuss the efficacy of those strategies.

MP.6 Attend to precision. Students attend to precision when they use place value language to explain their math drawings and calculations. They articulate the arithmetic properties they use to solve a variety of problems. For example, when adding 825 + 80, a student may show understanding of the associative property by saying, “I know that 20 + 80 equals 100, so I added 800 + 100 + 5, which equals 905.”

MP.7 Look for and make use of structure. Students look for and make use of the base ten structure when composing and decomposing. They extend their understanding from Module 4, viewing 10 tens as forming a new unit called a hundred, just as they understand that 10 ones forms 1 ten. They apply this understanding of base ten structure when adding and subtracting three-digit numbers, repeatedly bundling and unbundling groups of ten. They also make use of structure when they use simplifying strategies, such as compensation, to create a multiple of ten or a hundred.

MP.8 Look for and express regularity in repeated reasoning. As students repeatedly manipulate models and record the work abstractly, they recognize the cyclic pattern of the addition or subtraction of like units and the subsequent potential composition or decomposition of units through the place values. They see that the written form represents the same cycle they use with the manipulatives.

Term 3 Module 6: Focus Standards for Mathematical Practice

MP.3 Construct viable arguments and critique the reasoning of others. Students explain their thinking using drawings, models, and equations to lay the conceptual foundation for multiplication and division. “If I build an array with 3 columns of 4 objects, then I must have twelve objects, because 4 + 4 + 4 = 12. Likewise, if I partition my rectangle into twelve equally sized tiles, I can make 3 equal groups of 4 tiles, or I can make 4 equal groups of 3 tiles.” Students also defend their reasoning as they prove that a number is even or odd, making connections to the previous concepts of counting by twos, adding on, equal groups, and doubles.

MP.4 Model with mathematics. Students learn to organize a set of objects into equal groups and then into rows and columns, or rectangular arrays. They use math drawings to analyze the relationship between rows and columns (e.g., 3 rows of 4, or 4 columns of 3) and to model the array as the sum of equal addends (e.g., 4 + 4 + 4 = 12).

MP.7 Look for and make use of structure. As students compose and decompose arrays, they recognize that the array structure is a collection of rows or columns and that either can be seen as a unit. Students match repeated addition to both the structure of the rows and columns, e.g., 5 + 5 + 5 can be 3 rows or columns of 5, or 3 fives.

MP.8 Look for and express regularity in repeated reasoning. As students create equal groups using objects, they recognize that they are repeatedly adding the same number, e.g., 3 groups of 4 bears can be expressed as 4 + 4 + 4. Students also discover patterns in odd and even numbers, recognizing the repetition of 0, 2, 4, 6, and 8 in the ones place.

Module 8: Time, Shapes, and Fractions as Equal Parts of Shapes

Work with time and money. (Focus on time. Money is addressed in Module 7. )

2.MD.7 Tell time and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

Reason with shapes and their attributes.

2.G.1 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. (Sizes are compared directly or visually, not compared by measuring.) Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Term 4 Module 7: Focus Standards for Mathematical Practice

MP.1 Make sense of problems and persevere in solving them. Students draw to determine the part–whole relationships embedded within various word problem types, and based on their analysis, they persevere to use various addition and subtraction strategies to solve problems. They then persist in making a statement of the solution to answer the question in the original context. In this module, the problem solving contexts involve length, money, and data.

MP.2 Reason abstractly and quantitatively. Students compare measurements using rulers, tape diagrams, and graphs. After they abstract the number of units or length measurements to calculate differences, they reinterpret the difference using the given units within a problem. Students also abstract the value from a set of coins to find the total value and then express that value once again in terms of dollars or cents.

MP.4 Model with mathematics. Students create drawings (e.g., tape diagrams) and write equations to model and solve word problems involving units of length, money, and data. Students use appropriate representations (e.g., line plot, bar graph, and picture graph) to visually display data. Students also use the number line to understand numbers and their relationships and to represent sums and differences within 100. Students organize their thinking about money by modeling with dollars and coins to solve addition and subtraction word problems.

MP.5 Use appropriate tools strategically. Students apply their measurement skills and knowledge of the ruler to measure a variety of objects using the appropriate measurement tools, such as inch rulers and yardsticks. When conventional measurement tools are not available, students make decisions about which resources might be helpful, such as using iteration with a shoe, a book, or a lima bean, while recognizing the limitations of such tools.

MP.6 Attend to precision. Students attend to precision when they iterate a physical unit to create inch rulers. They align the zero point on a ruler as the beginning of the total length, and they use various measurement tools and precise language to describe their experience: “I used an inch as the length unit.” Students learn estimation strategies for measurement and make closer and closer approximations to the actual length. They assign specific values to different coins and count up starting with the largest value. Students generate and represent data in a bar graph, picture graph, or line plot, labelling axes appropriately and specifying the unit of measure.

Term 4 Module 8: Focus Standards for Mathematical Practice

MP.1 Make sense of problems and persevere in solving them. Students are encouraged to persevere when arranging shapes to create specific composite shapes, when recomposing the pieces into different shapes, and when creating even larger shapes from composite shapes. When students partition composite shapes (e.g., circles and rectangles) into equal shares, they ask themselves, “How can I look at this differently?” Students organize their thinking through drawing, and they see, for example, that a circle can be described in terms of halves, thirds, or fourths.

MP.3 Construct viable arguments and critique the reasoning of others. Students use drawings and precise language to describe and analyze shapes, and they defend their reasoning as to what makes a quadrilateral, for example, a rhombus. Students also discuss the partitioning of a composite shape (e.g., a hexagon) and relate the different parts, or shares, to halves, thirds, and fourths. They make connections between fraction concepts and telling time, explaining the connection between their work with halves and quarters to the analog clock.

MP.6 Attend to precision. Students describe and analyze various two-dimensional shapes by attending to their specific attributes. Students accurately draw shapes using their knowledge of attributes and rulers. Then, while working with a partner, students name and analyze their partner’s shape drawings by counting the number of sides or angles. Students also appropriately name parts of a whole using terms such as halves, thirds, and fourths or quarters.

MP.7 Look for and make use of structure. Students identify attributes, such as the number of sides and angles, in order to classify shapes such as triangles and quadrilaterals. They make use of the part–whole structure to understand that a whole unit can be partitioned into equal shares, or smaller units (e.g., each of 4 equal shares = a fourth of the whole). Students use their understanding of the partitioning of a circle to tell time to the quarter and half hour. Through previous fluency practice, students use the pattern of skip-counting by fives to tell time on an analog clock.