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Mathematics | Grade 2 In Grade 2, instructional time should focus on four critical areas: (1) extending understanding of base-ten notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; and (4) describing and analyzing shapes.

(1) Students extend their understanding of the base-ten system. This includes ideas of counting in fives, tens, and multiples of hundreds, tens, and ones, as well as number relationships involving these units, including comparing. Students understand multi-digit numbers (up to 1000) written in base-ten notation, recognizing that the digits in each place represent amounts of thousands, hundreds, tens, or ones (e.g., 853 is 8 hundreds + 5tens + 3 ones).

(2) Students use their understanding of addition to develop fluency with addition and subtraction within 100. They solve problems within 1000 by applying their understanding of models for addition and subtraction, and they develop, discuss, and use efficient, accurate, and generalizable methods to compute sums and differences of whole numbers in base-ten notation, using their understanding of place value and the properties of operations. They select and accurately apply methods that are appropriate for the context and the numbers involved to mentally calculate sums and differences for numbers with only tens or only hundreds.

(3) Students recognize the need for standard units of measure (centimeter and inch) and they use rulers and other measurement tools with the understanding that linear measure involves an iteration of units. They recognize that the smaller the unit, the more iterations they need to cover a given length.

(4) Students describe and analyze shapes by examining their sides and angles. Students investigate, describe, and reason about decomposing and combining shapes to make other shapes. Through building, drawing, and analyzing two- and three-dimensional shapes, students develop a foundation for understanding area, volume, congruence, similarity, and symmetry in later grades.

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Contents Mathematics | Grade 2 ................................................................................................................................. 1

Grade 2 Overview ......................................................................................................................................... 3

Operations and Algebraic Thinking 2.OA .................................................................... 4

Represent and solve problems involving addition and subtraction. (2.OA.A)...................................... 4 Add and subtract within 20. (2.OA.B) ................................................................................................... 5 Work with equal groups of objects to gain foundations for multiplication. (2.OA.C) .......................... 7

Number and Operations in Base Ten 2.NBT ............................................................... 13

Understand place value. (2.NBT.A) ..................................................................................................... 13 Use place value understanding and properties of operations to add and subtract. (2.NBT.B).......... 24

Measurement and Data 2.MD ............................................................... 35

Measure and estimate lengths in standard units. (2.MD.A)............................................................... 35 Relate addition and subtraction to length. (2.MD.B) ......................................................................... 41 Work with time and money. (2.MD.C) ................................................................................................ 42 Represent and interpret data. (2.MD.D)............................................................................................. 48

Geometry 2.G .................................................................. 52

Reason with shapes and their attributes. (2.G.A) ............................................................................... 52

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Grade 2 Overview Operations and Algebraic Thinking

• Represent and solve problems involving addition and subtraction.

• Add and subtract within 20. • Work with equal groups of objects to gain

foundations for multiplication. Number and Operations in Base Ten

• Understand place value. • Use place value understanding and properties of

operations to add and subtract. Measurement and Data

• Measure and estimate lengths in standard units. • Relate addition and subtraction to length. • Work with time and money. • Represent and interpret data.

Geometry

• Reason with shapes and their attributes.

Mathematical Practices 1. Make sense of problems and

persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

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Operations and Algebraic Thinking 2.OA

Represent and solve problems involving addition and subtraction. (2.OA.A) 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 (2.OA.A.1) (DOK 2)

a. Example: Solution (DOK 1)

b. Example: Solution (DOK 2)

c. Example: (Former NAEP question) (DOK 1)

Answer: B

1 See Glossary, Table 1.

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d. Example: (Former NAEP question) (DOK 1)

The weights on the scale above are balanced. Each cube weighs 3 pounds. The cylinder weighs N pounds. Which number sentence best describes this situation?

1. 6 + N = 12 2. 6 + N = 4 3. 2 + N = 12 4. 2 + N =4 Answer:1. 6 + N = 12

e. Example: (Former NAEP question) (DOK 1)

Paco had 32 trading cards. He gave N trading cards to his friend. Which expression tells how many trading cards Paco has now? 1. 32 + N 2. 32 – N 3. N – 32 4. 32 ÷ N Answer: 2. 32 - N

f. Example: (Former NAEP question) (DOK 1)

The numbers in the pattern below are increasing by 12. Which of these numbers is part of the pattern? 14, 26, 38, ___, ___

1. 52 2. 58 3. 60 4. 62 Answer: 4. 62

g. Example: (Former NAEP question) (DOK 1) What number should go in the blank above to make the number sentence true? 58 = ___ + 36 Answer:22

Add and subtract within 20. (2.OA.B)

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2. Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers. (2.OA.B.2) (DOK 1)

a. Example: Solution (DOK 1)

b. Example: Solution (DOK 2)

2 See standard 1.OA.6 for a list of mental strategies.

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Work with equal groups of objects to gain foundations for multiplication. (2.OA.C) 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.C.3) (DOK 2)

a. Example: Solution (DOK 3)

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b. Example: Solution (DOK 1)

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c. Example: (Former NAEP question) (DOK 3) Sam did the following problems. 2 + 1 = 3 6 + 1 = 7 Sam concluded that when he adds one to any whole number, his answer will always be odd. Is Same correct? Explain your answer. Answer: No: add a 1 to an odd whole number and it is even

d. Example: (Former NAEP question) (DOK 1) Write each of the following numbers in the circle where it belongs.

Answer: Odd (47), Even (30, 124)

e. Example: (Former NAEP question) (DOK 3) In each class listed below, the students are lining up with a partner to walk to lunch. Which class will have one child with no other child for a partner? Explain your choice.

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Answer: Mr. West’s Class because there is an odd number of students and 25 cannot be divided by 2.

f. Example: (Former NAEP question) (DOK 1) Which of these numbers is an odd number? 1. 6 2. 52 3. 111 4. 320 5. 536 Answer: 3. 111

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. (2.OA.C.4) (DOK 2)

a. Example: Solution (DOK 1)

b. Example: Solution (DOK 2)

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Number and Operations in Base Ten 2.NBT

Understand place value. (2.NBT.A) 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.A.1) (DOK 2) 1. Example: Solution (DOK 1)

2. Example: Solution (DOK 1)

3. Example: Solution (DOK 3)

4. Example: Solution (DOK 1)

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5. Example: Solution (DOK 2)

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6. Example: Solution (DOK 3)

7. Example: Solution (DOK 3)

8. Example: Solution (DOK 2)

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9. Example: Solution (DOK 2)

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10. Example: Solution (DOK 3)

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11. Example: Solution (DOK 2)

2. Count within 1000; skip-count by 5s, 10s, and 100s. (2.NBT.A.2) (DOK 1) a. Example: Solution (DOK 2)

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b. Example: (Former NAEP question) (DOK 1)

Answer: A

3. Read and write numbers to 1000 using base-ten numerals, number names, and expanded form. (2.NBT.A.3) (DOK 1,2)

a. Example: Solution (DOK 2)

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4. Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and

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< symbols to record the results of comparisons. (2.NBT.A.4) (DOK 2) a. Example: Solution (DOK 1)

b. Example: Solution (DOK 1)

c. Example: Solution (DOK 2)

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d. Example: Solution (DOK 1)

e. Example: Solution (DOK 1)

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f. Example: Solution (DOK 3)

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Use place value understanding and properties of operations to add and subtract. (2.NBT.B) 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.B.5) (DOK 1,2) a. Example: Solution (DOK 2)

b. Example: Solution (DOK 3)

c. Example: Solution (DOK 2)

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d. Example: (Former NAEP question) (DOK 2) While Adisha’s parents were looking for a car, Adisha counted the number of cards and trucks in the lot of the sales office. She counted: 25 new cars 16 used cars 59 trucks How many more trucks than cars are there on the lot? Write directions for how to use the calculator to solve this problem. Answer: There are 18 more trucks than cars on the lot. You would take 59 -16 – 25.

e. Example: (Former NAEP question) (DOK 2) Tanika wrote 100 in four different ways. 85 + 15 141 – 41 70 + 30 102 – 2 Write 100 in four other ways. Do not use the numbers Tanika used. 1. ______________ 2. ______________ 3. ______________ 4. ______________ Answer: 1. 25 + 75 2. 125 - 25 3. 97 + 3 4. 104 - 4

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

a. Example: Solution (DOK 2)

b. Example: (Former NAEP question) (DOK 1)

Add: 38 74 66 +75 Answer: 253

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.B.7) (DOK 2)

a. Example: Solution (DOK 3)

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b. Example: Solution (DOK 2)

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c. Example: (Former NAEP question) (DOK 3)

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Answer: 315 for Jill, Jill could win the game, Kevin could win the game, Ruth could not win the game. The highest number of points Kevin can get is 324. The highest amount Ruth can get is 306.

d. Example: (Former NAEP question) (DOK 1) 301 -75 1. 226 2. 235 3. 236 4. 374

Answer: 226

e. Example: (Former NAEP question) (DOK 1)

On Saturday 789 people went to the zoo. On Sunday 983 people went to the zoo. How many more people went to the zoo on Sunday than on Saturday? 1. 194 2. 204 3. 206 4. 1,722

Answer: 1. 194

f. Example: (Former NAEP question) (DOK 1)

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Answer: C. 251

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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.B.8) (DOK 2)

a. Example: Solution (DOK 3)

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9. Explain why addition and subtraction strategies work, using place value and the properties of operations.3

(2.NBT.B.9) (DOK 3) a. Example: Solution (DOK 3)

3 Explanations may be supported by drawings or objects.

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Measurement and Data 2.MD

Measure and estimate lengths in standard units. (2.MD.A) 1. Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter

sticks, and measuring tapes. (2.MD.A.1) (DOK 1) a. Example: Solution (DOK 3)

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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.A.2) (DOK 2,3) 3. Estimate lengths using units of inches, feet, centimeters, and meters. (2.MD.A.3) (DOK 2)

a. Example: Solution (DOK 3)

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4. Measure to determine how much longer one object is than another, expressing the length difference in

terms of a standard length unit. (2.MD.A.4) (DOK 1,2) a. Example: Solution (DOK 3)

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Relate addition and subtraction to length. (2.MD.B) b. 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.B.5) (DOK 2) 1. Example: Solution (DOK 2)

c. 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. (2.MD.B.6) (DOK 1,2) 1. Example: Solution (DOK 2)

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Work with time and money. (2.MD.C) d. Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

(2.MD.C.7) (DOK 1) 1. Example: Solution (DOK 2)

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IA.1.Describe the relationship among standard units of time: minutes, hours, days, weeks, months and years.

(2.MD.C.IA.1) (DOK 2,3) e. 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? (2.MD.C.8) (DOK 2) 1. Example: Solution (DOK 2)

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2. Example: Solution (DOK 3)

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3. Example: Solution (DOK 3)

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4. Example: Solution (DOK 3)

5. Example: Solution (DOK 3)

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6. Example: Solution (DOK 3)

7. Example: Solution (DOK 2)

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Represent and interpret data. (2.MD.D) f. Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by

making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units. (2.MD.D.9) (DOK 2) 1. Example: Solution (DOK 2)

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2. Example: Solution (DOK 2)

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3. Example: Solution (DOK 2)

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IA.2Use interviews, surveys, and observations to collect data that answer questions about students' interests and/o their environment. (2.MD.D.IA.2) (DOK 2,3) g. Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four

categories. Solve simple put-together, take-apart, and compare problems4 using information presented in a bar graph. (2.MD.D.10) (DOK 2) 1. Example: Solution (DOK 2)

4 See Glossary, Table 1.

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Geometry 2.G

Reason with shapes and their attributes. (2.G.A) 1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number

of equal faces.5 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. (2.G.A.1) (DOK 1,2) a. Example: Solution (DOK 2)

5 Sizes are compared directly or visually, not compared by measuring.

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b. Example: (Former NAEP question) (DOK 1)

What shapes make up the faces of a square pyramid?

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1. Triangles only 2. Pentagon and triangles 3. Square and rectangles 4. Square and triangles

Answer: 4. Square and triangles

c. Example: (Former NAEP question) (DOK 1)

What is the shape of the shaded figure inside the star?

1. Hexagon 2. Pentagon 3. Quadrilateral 4. Triangle Answer: 2. Pentagon

d. Example: (Former NAEP question) (DOK 1)

In the figure below, outline a four-sided shape that is not a rectangle (or a square).

Answer:

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e. Example: (Former NAEP question) (DOK 1) How many sides does a pentagon have? 1. Three 2. Four 3. Five 4. Six 5. Seven

Answer: 3. Five

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2. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. (2.G.A.2) (DOK 2) a. Example: Solution (DOK 2)

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. (2.G.A.3) (DOK 2,3) a. Example: Solution (DOK 3)

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