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.

4.1.6 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

 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

4.1.6 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

 2.         Construct, use, and explain procedures for performing calculations with fractions and decimals with:

·        Pencil-and-paper

·        Mental math

·        Calculator

Sample ECR Item: Jan brought eight 2-liter bottles of soda to the class party. At the end of the party, one bottle was ½ full, a second bottle contained 0.5 liters of soda, and a third bottle was 3/5 full. The other 5 bottles were empty. How much soda did the students drink during the class party?• Show one way to get the answer to this problem. Explain your method.• Show another way to get the answer to this problem. Explain your method.

Sample MC Item: Janis surveyed the students in her class and discovered that 2/3 of the class rides bicycles. There are 24 students in the class. How many of them ride bicycles?
a. 12

* b. 16

c. 18

d. 20

Sample SC Item: Sandra's dad works in a neighborhood pizza shop. He brought 6 ½ pizzas to Sandra's girl scout meeting on Tuesday evening. If each girl ate ¼ of a pizza, how many girls could be fed with the 6 ½ pizzas? (Answer: 26 girls)

Instructional/Assessment Focus:
• This is an area of focus in grade 5, but application to decimals is in grade 6.

Sample SCR Item: Sixteen students decide to share the cost of a DVD rental for a party. The DVD rental is $5.76. How much will each of them have to pay? (Answer: 36¢ or $0.36)

Sample SCR Item: Irma has $10.00 to spend on pencils. Each pencil costs $.40. How many pencils can she buy? (Answer: 25 pencils)

Sample MC Item: Which of the following numbers cannot be the area of a square whose sides have lengths given in whole numbers?

a. 25

* b. 84

c. 169

d. 196

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

 7.   Understand and use the various relationships among operations and properties of operations.

 8.      Understand and apply the standard algebraic order of operations for the four basic operations, including appropriate use of parentheses.

Sample MC Item: Evaluate 3 + 2 x 4.

a. 24

b. 20

* c. 11

d. 9

4.1.6 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

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.

4.2.6 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

 1.         Understand and apply concepts involving lines and angles.

·        Notation for line, ray, angle, line segment

·        Properties of parallel, perpendicular, and intersecting lines

·        Sum of the measures of the interior angles of a triangle is 180°

 2.         Identify, describe, compare, and classify polygons and circles.

·        Triangles by angles and sides

·        Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi

·        Polygons by number of sides.

·        Equilateral, equiangular, regular

·        All points equidistant from a given point form a circle

4.         Understand and apply the concepts of congruence and symmetry (line and rotational).

4.2.6 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

4.2.6 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

4.2.6 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

4.2.6 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

2.      Develop and apply strategies and formulas for finding perimeter and area.

·        Triangle, square, rectangle, parallelogram, and trapezoid

·        Circumference and area of a circle

Sample MC Item: The area of the base of a cereal box is 12 square inches. The box is 10 inches high. What is its volume?
* a. 120 cu. in.

b. 60 cu. in.

c. 40 cu. in.

d. 22 cu. in.

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.

4.3.6 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

 1.         Recognize, describe, extend, and create patterns involving whole numbers and rational numbers.

·        Descriptions using tables, verbal rules, simple equations, and graphs

·        Formal iterative formulas (e.g., NEXT = NOW * 3)

·        Recursive patterns, including Pascal’s Triangle (where each entry is the sum of the entries above it) and the Fibonacci Sequence: 1,  1,  2,  3,  5,  8, . . .    (where NEXT = NOW + PREVIOUS)

Sample MC Item: Which equation fits this pattern? 2, 6, 18, 24, . . a. NEXT = NOW + 4

b. NEXT = NOW + 3 *

c. NEXT = 3 * NOW

d. NEXT = NOW / 3

4.3.6 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 alge