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

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

1.         Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 2 pertain to these sets of numbers as well).

·        Whole numbers through hundreds

·        Ordinals

·        Proper fractions (denominators of 2, 3, 4, 8, 10)

4.         Count and perform simple computations with coins.

·        Amounts up to $1.00 (using cents notation)

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.

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

 5.         Understand the various uses of numbers.

·        Counting, measuring, labeling (e.g., numbers on baseball uniforms)

1.        Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 4 pertain to these sets of numbers as well).

·        Whole numbers through millions

·        Commonly used fractions (denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 16) as part of a whole, as a subset of a set, and as a location on a number line

·        Decimals through hundredths

Instructional/Assessment Focus:
• It is important to note that the sets of numbers specified in this CPI also apply to the other grade 4 mathematics CPIs (e.g., 4.1.4A3 and 4.1.4A6 below).

Sample Assessment Item:
• Extended Constructed Response (ECR): A class of 24 students will perform an act for the spring talent show. In the class, 2/3 of the students want to perform a skit. The rest of the students want to sing a song. The teacher decided that 3/4 of the students must agree on an act before the decision will be final.
• How many of the students want to perform a skit?
• How many more students would have to choose a skit before 3/4 of the students agree on it?
• Show all of your work and explain your answer.
(Note: Students may draw a picture in response to this question; they are not expected to use formal algorithms for working with fractions at this grade level.)

2.         Demonstrate an understanding of place value concepts.

Sample Assessment Item:
Multiple Choice (MC): Using the digits 1 - 7 only once, what is the largest even number you can make with a 5 in the thousands place?
a. 7,654,321

b. 7,654,312

* c. 7,645,312

d. 7,435,216

3.         Demonstrate a sense of the relative magnitudes of numbers.

Instructional/Assessment Focus:
•Refers not only to whole numbers, but also to fractions and decimals, as specified in 4.1.4A1.

Sample Assessment Item:
MC: If the following fractions were graphed on a number line, which fraction would be closest to zero?
a. 2/3

b. 1/4

c. 3/8

d. 1/10

4.         Understand the various uses of numbers.

·        Counting, measuring, labeling (e.g., numbers on baseball uniforms), locating (e.g., Room 235 is on the second floor)

5.         Use concrete and pictorial models to relate whole numbers, commonly used fractions, and decimals to each other, and to represent equivalent forms of the same number.

Sample Assessment Item:
• SCR: How many wholes are there in 16/8? ________
(This item would appear on a non-calculator portion of the statewide assessment. Answer: two or 2)

6.         Compare and order numbers.

Instructional/Assessment Focus:
• Refers not only to whole numbers, but also to fractions and decimals, as specified in 4.1.4A1.

Sample Assessment Item:
MC: Which of the following shows the decimals in order from least to greatest?
a. 0.5, 0.45, 0.54
* b. 0.45, 0.5, 0.54
c. 0.54, 0.5, 0.45
d. 0.45, 0.54,  0.5

7.         Explore settings that give rise to negative numbers.

·        Temperatures below 0^o, debts

·        Extension of the number line

Instructional/Assessment Focus:
• Students should have the opportunity to explore settings that give rise to negative numbers (e.g., temperatures below 0o, debts, games that involve negative numbers). This would include the use of a thermometer in science experiments.
• This content should be introduced at this grade level, but mastery of the content is not assessed in statewide assessment at this grade level.

1.         Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 5 pertain to these sets of numbers as well): All fractions as part of a whole, as subset of a set, as a location on a number line, and as divisions of whole numbers; All decimals.

Sample Short Constructed Response (SCR) Item: Four friends have three brownies left over from a party. They would like to split them equally. How much should each of them receive? (Answer: 75% or .75 or 3/4 of a brownie)

Assessment Focus:
• The emphasis in statewide assessment is on the computation.

Sample Multiple Choice (MC) Item: Debbie has a $5 bill. She wants to purchase a notebook for 75¢ and a pen for 50¢. How much money will Debbie have left after purchasing the notebook and the pen?
a. $1.25 b. $2.75 * c. $3.75 d. $4.25

Sample Short Constructed Response (SCR) Item: Juliette has a $5 bill. She wants to purchase a notebook for 75¢ and a pen for 50¢. How much money will Juliette have left after purchasing the notebook and the pen? (Answer: $3.75)

Instructional/Assessment Focus:
• Refers not only to whole numbers, but also to fractions and decimals, as specified in 4.1.5A1.

Sample MC Item: If these fractions were graphed on the number line, which of them would be closest to zero?
a. 3/5

b. 1/4

c. 3/20

* d. 1/10

Sample MC Item: Which of the following is equivalent to 3/4?
a. .25

b. 4/3

c. .85

* d. 9/12

Assessment Focus:
• The emphasis in statewide assessment is on application.

Sample MC Item: How many numbers between 20 and 50 have no remainder when divided by 6?
a. 3

b. 4

* c. 5

d. 6

Instructional/Assessment Focus:
• Refers not only to whole numbers, but also to fractions and decimals, as specified in 4.1.5A1.

Sample SCR Item: State a number that is between 1/3 and 0.36.

Acceptable answers would include various representations of Real Numbers between 1/3 and .36 (e.g., 0.34, 0.334, 0.35, 7/20, etc.)

Sample Extended Constructed Response (ECR) Item: On the number line in your answer folder, plot points for the following numbers.
4/5, 0.6
• Label each point.
• Name two different rational numbers that are greater than 0.6 and less than 4/5. (Write one of your numbers in fractional form and write the other number in decimal form.)
• Explain how you know that each of your numbers is greater than 0.6 and less than 4/5.

 1.         Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 6 pertain to these sets of numbers as well).

·        All integers

·        All fractions as part of a whole, as subset of a set, as a location on a number line, and as divisions of whole numbers

·        All decimals

This is an area of focus in grade 5 and may be assessed at a higher level of understanding in grade 6.

Sample Extended Constructed Response (ECR) Item: Notebooks at the school store cost 75¢ each. Pens cost 50¢ each. How many different combinations of notebooks and pens could Hermit buy for $5.00? Explain your reasoning.

Sample Short Constructed Response (SCR) Item: Yusuke has a $5 bill. He wants to purchase 3 notebooks, for 75¢ each. How much money will Yusuke have left after purchasing the 3 notebooks? (Answer: $2.75)

 Sample Multiple Choice (MC) Item: Tim has a $5 bill. He wants to purchase 3 notebooks, for 75¢ each. How much money will Tim have left after purchasing the notebooks?

a. $2.25

* b. $2.75

c. $3.75

d. $4.25

Instructional/Assessment Focus:
• Includes, for example, the recognition that when adding one hundred and one million, the answer would be very close to one million.

Instructional 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 and use whole-number percents between 1 and 100 in a variety of situations.

This is an area of focus in grade 5 and may be assessed at a higher level of understanding in grade 6.

7.     Develop and apply number theory concepts in problem solving situations.

·        Primes, factors, multiples

·        Common multiples, common factors

·       Least common multiple, greatest common factor

The third bullet of this CPI was added by the State Board of Education on January 9, 2008.

8.         Compare and order numbers.

Instructional/Assessment Focus:
• Refers to integers, fractions, and decimals, as specified in 4.1.6A1; and
• Students might be asked to put numbers (including fractions and decimals) in order from least to greatest.

Sample MC Item: The table below shows the low temperatures of four New Jersey Cities on one winter night.

Which city had the lowest temperature that night?

a. Gloucester

b. New Brunswick

* c. Elizabeth

d. Paterson

1.         Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 7 pertain to these sets of numbers as well):

·        Rational numbers

·        Percents

·        Whole numbers with exponents

Instructional/Assessment Focus:

Refers to Rational numbers; Percents; and Whole numbers with exponents, as specified in 4.1.7A1

Sample Multiple Choice (MC) Item: A carpenter wants to drill a hole that is just slightly larger than ¼ inch in diameter. Which of these is the smallest, but still greater than ¼ inch?
a. 3/16 inch

b. 7/32 inch

c. 5/16 inch

* d. 9/32 inch

Assessment of this CPI is generally within the context of one or more of the other content CPIs.

1.         Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 8 pertain to these sets of numbers as well):

·        Rational numbers

·        Percents

·        Exponents

·        Roots

·        Absolute values

·        Numbers represented in scientific notation

3.         Understand and use ratios, rates,  proportions, and percents (including percents greater than 100 and less than 1) in a variety of situations.

• The word “rates” was added to this CPI by the State Board of Education on January 9, 2008. This is an area of focus in grade 8 and should be linked to the concept of slope (4.3.8B1).

6.         Recognize that repeating decimals correspond to fractions and determine their fractional equivalents.

·        5/7 = 0. 714285714285…  =  0.

Sample ECR Item: With only a DVD and a piece of string, explain how you could approximate the value of π.

4.1 B. Numerical Operations

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

1.         Develop the meanings of addition and subtraction by concretely modeling and discussing a large variety of problems.

·        Joining, separating, and comparing

4.         Construct, use, and explain procedures for performing addition and subtraction calculations with:

·        Pencil-and-paper

·        Mental math

·        Calculator

5.         Use efficient and accurate pencil-and-paper procedures for computation with whole numbers.

·        Addition of 2-digit numbers

·        Subtraction of 2-digit numbers

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