STANDARD 4.1: 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.

Link to Standard 4.1 Grade K-2

Link to Standard 4.1 Grade 4

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