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.

4.1.7 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.         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.

4.1.7 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

1.         Use and explain procedures for performing calculations with integers and all number types named above with:

·        Pencil-and-paper

·        Mental math

·        Calculator

Sample MC Item: A used car was priced at $7000. The salesperson then offered a discount of $350. This discount represented what percent off the original price?

* a. 5

b. 20

c. 80

d. 95

4.1.7 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

1.    Use equivalent representations of numbers such as fractions, decimals, and percents to facilitate estimation.

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.

4.2.7 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 properties of polygons.

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

·        Regular polygons

2.         Understand and apply the concept of similarity.

·        Using proportions to find missing measures

·        Scale drawings

·        Models of 3D objects

3.         Use logic and reasoning to make and support conjectures about geometric objects.

4.2.7 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

1.         Understand and apply transformations.

·        Finding the image, given the pre-image, and vice-versa

·        Sequence of transformations needed to map one figure onto another

·        Reflections, rotations, and translations result in images congruent to the pre-image

·        Dilations (stretching/shrinking) result in images similar to the pre-image

4.2.7 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

1.         Use coordinates in four quadrants to represent geometric concepts.

4.2.7 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

Sample MC Item: Mr. Hernandez wants to tile his table-top with rectangular tiles, each measuring 3 inches by 4 inches. Which of the following represents the least number of tiles he can use if the table-top is a square measuring three feet on each side?

a. 36

b. 72

* c. 108

d. 144

Sample MC Item: Beth is covering an 8-foot by 10-foot wall with cork tiles. Each tile is a 6-inch by 6-inch square. How many tiles will she need to cover the wall?

a. 80

b. 160

* c. 320

d. 640

4.2.7 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

1.         Develop and apply strategies for finding perimeter and area.

·        Geometric figures made by combining triangles, rectangles and circles or parts of circles

·        Estimation of area using grids of various sizes

Sample MC Item:

The cone below has the same height and same size base as the cylinder. The cone has a total volume of 12 cu. in. [Insert appropriate diagram] What is the volume of the cylinder?

a. 4 cu. in.

b. 12 cu. in.

c. 24 cu. in.

* d. 36 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.

4.3.7 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, rational numbers, and integers.

·        Descriptions using tables, verbal and symbolic rules, graphs, simple equations or expressions

·        Finite and infinite sequences

·        Generating sequences by using calculators to repeatedly apply a formula

4.3.7 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

 1.         Graph functions, and understand and describe their general behavior.

·        Equations involving two variables

4.3.7 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)

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

2.        Use patterns, relations, symbolic algebra, and linear functions to model situations.

·        Using manipulatives, tables, graphs, verbal rules, algebraic expressions/equations/inequalities

·        Growth situations, such as population growth and compound interest, using recursive (e.g., NOW-NEXT) formulas (cf. science standard 5.5 and social studies standard 6.6)

4.3.7 D. Procedures

Descriptive Statement: Techniques for manipulating algebraic expressions - procedures - remain important, especially for students who may continue their study of mathematics in a calculus program. Utilization of algebraic procedures includes understanding and applying properties of numbers and operations, using symbols and variables appropriately, working with expressions, equations, and inequalities, and solving equations and inequalities.

Essential Questions

Enduring Understandings

- What makes an algebraic algorithm both effective and efficient? (4.5D1)

-  Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole.

- Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra. (4.5D1; 4.5D3; 4.5D4; 4.5D5)

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

1.       Use graphing techniques on a number line.

·        Absolute value

·        Arithmetic operations represented by vectors (arrows) (e.g., “-3 + 6” is “left 3, right 6”)

 2.      Solve simple linear equations informally and graphically.

·        Multi-step, integer coefficients only (although answers may not be integers)

·        Using paper-and-pencil, calculators, graphing calculators, spreadsheets, and other technology

Use the following equation to determine h, the number of hours Monica must have worked in the garden. 8h + 12 = 104

((Answer: 11 1/2 hours)

3.       Create, evaluate, and simplify algebraic expressions involving variables.

·        Order of operations, including appropriate use of parentheses

·        Substitution of a number for a variable

 4.       Understand and apply the properties of operations, numbers, equations, and inequalities.

·        Additive inverse

·        Multiplicative inverse

Standard 4.4 Data Analysis, Probability, and Discrete Mathematics
All students will develop an understanding of the concepts and techniques of data analysis, probability, and discrete mathematics, and will use them to model situations, solve problems, and analyze and draw appropriate inferences from data.

Big Idea Data Analysis: Reading, understanding, interpreting, and communicating data are critical in modeling a variety of real-world situations, drawing appropriate inferences, making informed decisions, and justifying those decisions.
Big Idea Probability: Probability quantifies the likelihood that something will happen and enables us to make predictions and informed decisions.
Big Idea Discrete Mathematics: Discrete mathematics consists of tools and strategies for representing, organizing, and interpreting non-continuous data.

4.4.7 A. Data Analysis

Descriptive Statement: In today's information-based world, students need to be able to read, understand, and interpret data in order to make informed decisions. In the early grades, students should be involved in collecting and organizing data, and in presenting it using tables, charts, and graphs. As they progress, they should gather data using sampling, and should increasingly be expected to analyze and make inferences from data, as well as