STANDARD 4: Mathematics

Mathematics

Mission: Through mathematics, students communicate, make connections, reason, and represent the world quantitatively in order to pose and solve problems.

Standard 4.1 Number and Numerical Operations

All students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways.

Big Idea: Numeric reasoning involves fluency and facility with numbers.
4.1 A. Number Sense
Descriptive Statement: Number sense is an intuitive feel for numbers and a common sense approach to using them. It is a comfort with what numbers represent that comes from investigating their characteristics and using them in diverse situations. It involves an understanding of how different types of numbers, such as fractions and decimals, are related to each other, and how each can best be used to describe a particular situation. It subsumes the more traditional category of school mathematics curriculum called numeration and thus includes the important concepts of place value, number base, magnitude, and approximation and estimation.
Essential Questions	Enduring Understandings
- How do mathematical ideas interconnect and build on one another to produce a coherent whole? (4.5C1; 4.5C6) - How can we compare and contrast numbers? (4.5A4) - How can counting, measuring, or labeling help to make sense of the world around us?	- One representation may sometimes be more helpful than another; and, used together, multiple representations give a fuller understanding of a problem. - A quantity can be represented numerically in various ways. Problem solving depends upon choosing wise ways. - Numeric fluency includes both the understanding of and the ability to appropriately use numbers.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 3:
1. Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 3 pertain to these sets of numbers as well). · Whole numbers through hundred thousands · Commonly used fractions (denominators of 2, 3, 4, 5, 6, 8, 10) as part of a whole, as a subset of a set, and as a location on a number line	Instructional/Assessment Focus: • It is important to note that the sets of numbers specified in this CPI also apply to the other grade 3 mathematics CPIs, including, for example, 4.1.3A5 and 4.1.3B6.
2. Demonstrate an understanding of whole number place value concepts.	Sample Assessment Items: • Multiple Choice (MC): What is the value of the 3 in 75,314? a. thirty * b. three hundred c. three thousand d. thirty thousand •MC: Using the digits 1 - 5 only once, what is the largest even number you can make with a 5 in the hundreds place? a. 54,321 b. 54,312 * c. 43,512 d. 32,514
3. Identify whether any whole number is odd or even.	Suggested Instructional/Assessment Strategies: • Students read literature that incorporates basic number concepts in an enjoyable and engaging way (e.g., Even Steven and Odd Todd, a Hello Reader by Kathryn Cristaldi et al. Scholastic, Inc., 1996).
4. Explore the extension of the place value system to decimals through hundredths.	Instructional/Assessment Focus: • This content should be introduced at this grade level, but mastery of the content is not assessed in statewide assessment at this grade level.
5. Understand the various uses of numbers. · Counting, measuring, labeling (e.g., numbers on baseball uniforms)	Instructional/Assessment Focus: • Refers not only to whole through hundred thousands, but also commonly used fractions (denominators of 2, 3, 4, 5, 6, 8, 10), as specified in 4.1.3A1.
6. Compare and order numbers.	Instructional/Assessment Focus: • Refers not only to whole through hundred thousands, but also commonly used fractions (denominators of 2, 3, 4, 5, 6, 8, 10), as specified in 4.1.3A1.
4.1 B. Numerical Operations
Descriptive Statement: Numerical Operations are an essential part of the mathematics curriculum, especially in the elementary grades. Students must be able to select and apply various computational methods, including mental math, pencil-and-paper techniques, and the use of calculators. Students must understand how to add, subtract, multiply, and divide whole numbers, fractions, decimals, and other kinds of numbers. With the availability of calculators that perform these operations quickly and accurately, the instructional emphasis now is on understanding the meanings and uses of these operations, and on estimation and mental skills, rather than solely on the development of paper-and-pencil proficiency.
Essential Questions	Enduring Understandings
- What makes a computational strategy both effective and efficient? (4.5D1) - How do operations affect numbers? - How do mathematical representations reflect the needs of society across cultures? (An essential question with broad applicability across multiple standards) (4.5C5)	- Computational fluency includes understanding not only the meaning, but also the appropriate use of numerical operations. - The magnitude of numbers affects the outcome of operations on them. - In many cases, there are multiple algorithms for finding a mathematical solution, and those algorithms are frequently associated with different cultures.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 3:
1. Develop the meanings of the four basic arithmetic operations by modeling and discussing a large variety of problems. · Addition and subtraction: joining, separating, comparing · Multiplication: repeated addition, area/array · Division: repeated subtraction, sharing	Instructional/Assessment Focus: • The focus in grade 3 is on developing meanings for multiplication and division. Students should have developed meanings for addition and subtraction in grades 1 and 2.
2. Develop proficiency with basic multiplication and division number facts using a variety of fact strategies (such as “skip counting” and “repeated subtraction”).	Sample Assessment Item: • Short Constructed Response (SCR): Brett is taking care of his neighbor’s dog for 7 days. Brett needs to let the dog outside 3 times a day. In all, how many times will Brett let the dog out? (This item would appear on a non-calculator portion of the statewide assessment. Answer: 21 times or 21)
3. Construct, use, and explain procedures for performing whole number calculations with: · Pencil-and-paper · Mental math · Calculator	Sample Assessment Items: • MC: 376 + 119 + 85 = a. 460 * b. 580 c. 1,230 d. 1,345 (This item would appear on a non-calculator portion of the statewide assessment.) • SCR: Mark has a stamp collection. He has 22 stamps from Japan, 34 from Canada, and 17 from Mexico. How many stamps does he have in all? (This item would appear on a non-calculator portion of the statewide assessment. Answer: 73 stamps.)
4. Use efficient and accurate pencil-and-paper procedures for computation with whole numbers. · Addition of 3-digit numbers · Subtraction of 3-digit numbers · Multiplication of 2-digit numbers by 1-digit numbers	Sample Assessment Items: • MC: Find the exact answer: 110 marbles + 70 marbles = a. 18 marbles b. 81 marbles * c. 180 marbles d. 810 marbles (This item would appear on a non-calculator portion of the statewide assessment.) • MC: Find the exact value of 24 x 7 a. 31 b. 141 c. 148 * d. 168 (This item would appear on a non-calculator portion of the statewide assessment.) • SCR: Find the exact answer: 110 + 70 = ______ (This item would appear on a non-calculator portion of the statewide assessment. Answer: 180) • SCR: Find the exact answer: 145 + 281 + 62 = ______ (This item would appear on a non-calculator portion of the statewide assessment. Answer: 488) • SCR: John had 365 pennies. He gave 56 pennies to his sister. How many pennies does John have left? ______ (This item would appear on a non-calculator portion of the statewide assessment. Answer: 309 pennies or 309¢ or $3.09)
5. Count and perform simple computations with money. · Cents notation (¢)	Sample Assessment Items: • SCR: What is the fewest number of coins needed to make 99¢, using only pennies and dimes? (Answer: 18) • MC: What is the fewest number of coins needed to make 99¢, using only pennies and dimes? a. 8 b. 14 * c. 18 d. 19 • Extended Constructed Response (ECR): A juice machine charges 65¢ for a can of juice and accepts only nickels, dimes, and quarters. The machine requires exact change. • Show a combination of the exact number of coins you could put in the juice machine to get a can of juice. • Is there another combination of coins you could use to get a can of juice? Show your work or explain your answer.
6. Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.
7. Check the reasonableness of results of computations.	Suggested Instructional/Assessment Strategy: • Note the connection to Estimation CPI 4.1.3C4.
4.1 C. Estimation
Descriptive Statement: Estimation is a process that is used constantly by mathematically capable adults, and one that can be easily mastered by children. It involves an educated guess about a quantity or an intelligent prediction of the outcome of a computation. The growing use of calculators makes it more important than ever that students know when a computed answer is reasonable; the best way to make that determination is through the use of strong estimation skills. Equally important is an awareness of the many situations in which an approximate answer is as good as, or even preferable to, an exact one. Students can learn to make these judgments and use mathematics more powerfully as a result.
Essential Questions	Enduring Understandings
- How can we decide when to use an exact answer and when to use an estimate?	- Context is critical when using estimation.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 3:
1. Judge without counting whether a set of objects has less than, more than, or the same number of objects as a reference set.
2. Construct and use a variety of estimation strategies (e.g., rounding and mental math) for estimating both quantities and the result of computations.	Instructional/Assessment Focus: • Assessment of this CPI is frequently within the context of CPI 4.1.3C4 or one or more of the other content CPIs. • A common student mistake is rounding single-digit numbers (e.g., mentally converting 36 x 7 to 40 x 10 = 400, rather than to 40 x 7 = 280). Sample Assessment Items: • MC: Estimate 123 + 685. The sum is between which numbers? a. 400 and 600 b. 700 and 900 c. 1,000 and 1,200 d. 1,300 and 1,500 (This item would appear on a non-calculator portion of the statewide assessment.) • MC: Find the exact answer: 900 – 201 = a. 699 b. 700 c. 701 d. 799 (This item would appear on a non-calculator portion of the statewide assessment.) • MC: Estimate 423 - 174. The difference is between which numbers? a. 0 and 199 * b. 200 and 399 c. 400 and 599 d. 600 and 799 (This item would appear on a non-calculator portion of the statewide assessment.) • MC: Sandra traveled 458 miles to North Carolina, then 231 miles from North Carolina to West Virginia, and finally 340 miles home. Which of the following best describes the distance Sandra traveled? a. 600 mi b. 800 mi * c. 1000 mi d. 1200 mi
3. Recognize when an estimate is appropriate, and understand the usefulness of an estimate as distinct from an exact answer.	Instructional/Assessment Focus: • Assessment of this CPI and demonstration of this understanding is frequently within the context of one or more of the other content CPIs. • Student articulation of this understanding is expected to be evolving in grade 3. Statewide assessment of the concept should receive greater attention in later grades.
4. Use estimation to determine whether the result of a computation (either by calculator or by hand) is reasonable.	Sample Assessment Items: • ECR: Your friend Susan said that 454 + 42 = 432. Use estimation to explain why you think Susan is wrong. • ECR: Sam and Kelly were adding the numbers of students in their two schools. Sam told Kelly that 367 + 417 = 600. Use estimation to explain if you think Sam is right or wrong and why. • ECR: Peter discovered that the school enrollment this year is 150 less than last year, when there were 826 students. Kiesha told Peter that there are now about 575 students. Use estimation to explain why you think Kiesha is right or wrong.

Standard 4.2 Geometry and Measurement

All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena.

Big Idea Geometry: Spatial sense and geometric relationships are a means to solve problems and make sense of a variety of phenomena. Big Idea Measurement: Measurement is a tool to quantify a variety of phenomena.
4.2 A. Geometric Properties
Descriptive Statement: This includes identifying, describing and classifying standard geometric object, describing and comparing properties of geometric objects, making conjectures concerning them, and using reasoning and proof to verify or refute conjectures and theorems. Also included here are such concepts as symmetry, congruence, and similarity.
Essential Questions	Enduring Understandings
- How can spatial relationships be described by careful use of geometric language? - How do geometric relationships help in solving problems and/or make sense of phenomena?	- Geometric properties can be used to construct geometric figures. (4.5D1; 4.5D2; 4.5E3) - Geometric relationships provide a means to make sense of a variety of phenomena.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 3:
1. Identify and describe spatial relationships of two or more objects in space. · Direction, orientation, and perspectives (e.g., which object is on your left when you are standing here?) · Relative shapes and sizes
2. Use properties of standard three-dimensional and two-dimensional shapes to identify, classify, and describe them. · Vertex, edge, face, side, angle · 3D figures – cube, rectangular prism, sphere, cone, cylinder, and pyramid · 2D figures – square, rectangle, circle, triangle, pentagon, hexagon, octagon
3. Identify and describe relationships among two-dimensional shapes. · Same size, same shape · Lines of symmetry
4. Understand and apply concepts involving lines, angles, and circles. · Line, line segment, endpoint
5. Recognize, describe, extend, and create space-filling patterns.
4.2 B. Transforming Shapes
Descriptive Statement: This includes identifying, describing and classifying standard geometric object, describing and comparing properties of geometric objects, making conjectures concerning them, and using reasoning and proof to verify or refute conjectures and theorems. Also included here are such concepts as symmetry, congruence, and similarity.
Essential Questions	Enduring Understandings
- What situations can be analyzed using transformations and symmetries? (4.5E1; 4.5E2; 4.5E3)	- Shape and area can be conserved during mathematical transformations..
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 3:
1. Describe and use geometric transformations (slide, flip, turn).
2. Investigate the occurrence of geometry in nature and art.	Instructional/Assessment Focus: • This CPI is largely an instructional CPI. Assessment of this CPI is generally within the context of one or more of the other content CPIs.
4.2 C. Coordinate Geometry
Descriptive Statement: Coordinate geometry provides an important connection between geometry and algebra. It facilitates the visualization of algebraic relationships, as well as an analytical understanding of geometry.
Essential Questions	Enduring Understandings
- How can geometric/algebraic relationships best be represented and verified? (4.5C2; 4.5D2; 4.5E1; 4.5E2; 4.5F5)	- Reasoning and/or proof can be used to verify or refute conjectures or theorems in geometry (4.5D1; 4.5D3; 4.5D4; 4.5D5; 4.5F5) - Coordinate geometry can be used to represent and verify geometric/algebraic relationships.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 3:
1. Locate and name points in the first quadrant on a coordinate grid.
4.2 D. Units Of Measurement
Descriptive Statement: Measurement helps describe our world using numbers. An understanding of how we attach numbers to real-world phenomena, familiarity with common measurement units (e.g., inches, liters, and miles per hour), and a practical knowledge of measurement tools and techniques are critical for students' understanding of the world around them.
Essential Questions	Enduring Understandings
- How can measurements be used to solve problems? (4.5A6)	- Everyday objects have a variety of attributes, each of which can be measured in many ways. -What we measure affects how we measure it. (4.5A4; 4.5A6) - Measurements can be used to describe, compare, and make sense of phenomena.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 3:
1. Understand that everyday objects have a variety of attributes, each of which can be measured in many ways.
2. Select and use appropriate standard units of measure and measurement tools to solve real-life problems. · Length – fractions of an inch (1/4, 1/2), mile, decimeter, kilometer · Area – square inch, square centimeter · Weight – ounce · Capacity – fluid ounce, cup, gallon, milliliter
3. Incorporate estimation in measurement activities (e.g., estimate before measuring).

4.2 E. Measuring Geometric Objects
Descriptive Statement: This area focuses on applying the knowledge and understandings of units of measurement in order to actually perform measurement. While students will eventually apply formulas, it is important they develop and apply strategies that derive from their understanding of the attributes. In addition to measuring objects directly, students apply indirect measurement skills, using, for example, similar triangles and trigonometry.
Essential Questions	Enduring Understandings
- How can measurements be used to solve problems? (4.5A6)	- Everyday objects have a variety of attributes, each of which can be measured in many ways. - What we measure affects how we measure it. (4.5A4; 4.5A6)** - Measurements can be used to describe, compare, and make sense of phenomena.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 3:
1. Determine the area of simple two-dimensional shapes on a square grid.
2. Determine the perimeter of simple shapes by measuring all of the sides
3. Measure and compare the volume of three–dimensional objects using materials such as rice or cubes.	Instructional/Assessment Focus: • Students are expected to solve problems (4.5A2)** involving this recognition. • The emphasis in grade 3 would be on the “measure,” rather than the “compare.”

Standard 4.3 Patterns and Algebra
All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.

Big Idea Algebra provides language through which we communicate the patterns in mathematics.
4.3 A. Patterns
Descriptive Statement: Algebra provides the language through which we communicate the patterns in mathematics. From the earliest age, students should be encouraged to investigate the patterns that they find in numbers, shapes, and expressions, and by doing so, to make mathematical discoveries. They should have opportunities to analyze, extend, and create a variety of patterns and to use pattern-based thinking to understand and represent mathematical and other real-world phenomena.
Essential Questions	Enduring Understandings
- How can change be best represented mathematically? (4.5C1; 4.5F1; 4.5F2; 4.5F3; 4.5F4) - How can patterns, relations, and functions be used as tools to best describe and help explain real-life situations? (4.5C1)	- The symbolic language of algebra is used to communicate and generalize the patterns in mathematics. - Algebraic representation can be used to generalize patterns and relationships.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 3:
1. Recognize, describe, extend, and create patterns. · Descriptions using words and number sentences/expressions · Whole number patterns that grow or shrink as a result of repeatedly adding, subtracting, multiplying by, or dividing by a fixed number (e.g., 5, 8, 11, . . . or 800, 400, 200, . . .)
4.3 B. Functions and Relationships
Descriptive Statement: The function concept is one of the most fundamental unifying ideas of modern mathematics. Student begin their study of functions in the primary grades, as they observe and study patterns. As students grow and their ability to abstract matures, students form rules, display information in a table or chart, and write equations which express the relationships they have observed. In high school, they use the more formal language of algebra to describe these relationships.
Essential Questions	Enduring Understandings
- How are patterns of change related to the behavior of functions? (4.5F1; 4.5F2; 4.5F3; 4.5F4)	- Patterns and relationships can be represented graphically, numerically, symbolically, or verbally. (4.5E1)
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 3:
1. Use concrete and pictorial models to explore the basic concept of a function. · Input/output tables, T-charts
4.3 C. Modeling
Descriptive Statement: The function concept is one of the most fundamental unifying ideas of modern mathematics. Student begin their study of functions in the primary grades, as they observe and study patterns. As students grow and their ability to abstract matures, students form rules, display information in a table or chart, and write equations which express the relationships they have observed. In high school, they use the more formal language of algebra to describe these relationships.
Essential Questions	Enduring Understandings
- How are mathematical models used to describe physical relationships? (4.5E2) - How are physical models used to clarify mathematical relationships? (4.5E3)	- Mathematical models can be used to describe and quantify physical relationships. (4.5E2) - Physical models can be used to clarify mathematical relationships. (4.5E3)