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.

4.1.8 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 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.

4.1.8 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 involving addition, subtraction, multiplication, division, and exponentiation with integers and all number types named above with:

·        Pencil-and-paper

·        Mental math

·        Calculator

Pat has 1296 one-inch square tiles. Which of the following are the dimensions of the largest square table top Pat could cover with the tiles? a. 324 in. x 324 in.

b. 9 ft x 9 ft

* c. 1 yd x 1 yd

d. 36 m x 36 m

4.1.8 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.

4.2.8 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, angles, and planes.

·        Complementary and supplementary angles

·        Vertical angles

·        Bisectors and perpendicular bisectors

·        Parallel, perpendicular, and intersecting planes

·        Intersection of plane with cube, cylinder, cone, and sphere

3.         Understand and apply properties of polygons.

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

·        Regular polygons

·        Sum of measures of interior angles of a polygon

·        Which polygons can be used alone to generate a tessellation and why

·        Using proportions to find missing measures

·        Scale drawings

·        Models of 3D objects

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

4.2.8 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

 2.         Use iterative procedures to generate geometric patterns.

·        Fractals (e.g., the Koch Snowflake)

·        Self-similarity

·        Construction of initial stages

·        Patterns in successive stages (e.g., number of triangles in each stage of Sierpinski’s Triangle)

4.2.8 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.

 2.         Use a coordinate grid to model and quantify transformations (e.g., translate right 4 units).

4.2.8 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 Short Constructed Response (SCR) Item: A tire is 25 inches in diameter. How many times will it turn in traveling a mile?

 6.         Solve problems that involve compound measurement units, such as speed (miles per hour), air pressure (pounds per square inch), and population density (persons per square mile).

4.2.8 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

·        Impact of a dilation on the perimeter and area of a 2-dimensional figure

3.         Develop and apply strategies and formulas for finding the surface area and volume of a three-dimensional figure.

·        Volume - prism, cone, pyramid

·        Surface area - prism (triangular or rectangular base), pyramid (triangular or rectangular base)

·        Impact of a dilation on the surface area and volume of a three-dimensional figure

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.

4.3.8 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

·        Arithmetic sequences (i.e., sequences generated by repeated addition of a fixed number, positive or negative)

·        Geometric sequences (i.e., sequences generated by repeated multiplication by a fixed positive ratio, greater than 1 or less than 1)

·        Generating sequences by using calculators to repeatedly apply a formula

4.3.8 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)