Highlights of the New Learning for Grade-Eight Students Are

Grade 8

Overview

This overview provides only the highlights of the new learning that should take place at the seventh-grade level. The specific skills and subject matter that seventh graders should be taught in each of the five mathematical strands are set forth in the formal standards and indicators for these strands. To alert educators as to when the progression in learning should occur for students in this grade, specific language is used with certain indicators:

An indicator beginning with the phrase “Generate strategies” addresses a concept that is being formally introduced for the first time, and students must therefore be given experiences that foster conceptual understanding.
An indicator beginning with the phrase “Apply an algorithm,” “Apply a procedure,” “Apply procedures,” or “Apply formulas” addresses a concept that has been introduced in a previous grade: students should already have the conceptual understanding, and the goal must now be fluency.
An indicator beginning with the phrase “Apply strategies and formulas” or “Apply strategies and procedures” addresses a concept that is being formally introduced for the first time, yet the goal must nonetheless be that students progress to fluency.

Highlights of the new learning for grade-eight students are

applying an algorithm to add, subtract, multiply, and divide integers;
understanding the concept of irrational numbers;
applying procedures to approximate square and cube roots;
applying procedures to solve multistep equations;
classifying relationships between two variables as either linear or nonlinear;
identifying the coordinates of the x- and y-intercepts of a linear equation;
understanding slope as a constant rate of change;
applying the Pythagorean theorem;
using ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane;
applying a dilation on a square, rectangle, or right triangle in a coordinate plane and analyzing the effect;
applying strategies and formulas to determine volume of three-dimensional shapes;
using multistep unit analysis to convert between and with the U.S. Customary System and the metric system; and
applying procedures to compute the odds of a given event.

8-1

Grade 8

Mathematical Processes

The mathematical processes provide the framework for teaching, learning, and assessing in mathematics at all grade levels. Instructional programs should be built around these processes.

Standard 8-1:The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.

The indicators for this standard, which are appropriate for grades six through eight, are adapted from Principles and Standards for School Mathematics (NCTM 2000). Classroom application should be based on the standard and its indicators; the mathematical goals for the class; and the skills, needs, and understandings of the particular students.

Indicators

8-1.1Generate and solve complex abstract problems that involve modeling physical, social, or mathematical phenomena.

8-1.2Evaluate conjectures and pose follow-up questions to prove or disprove conjectures.

8-1.3Use inductive and deductive reasoning to formulate mathematical arguments.

8-1.4Understand equivalent symbolic expressions as distinct symbolic forms that represent the same relationship.

8-1.5Generalize mathematical statements based on inductive and deductive reasoning.

8-1.6Use correct and clearly written or spoken words, variables, and notations to communicate about significant mathematical tasks.

8-1.7Generalize connections among a variety of representational forms and real-world situations.

8-1.8Use standard and nonstandard representations to convey and support mathematical relationships.

Grade 8

Strand - Numbers and Operations

Big Ideas: Integers, Rational Numbers, Irrational Numbers, Cubes, Square Roots, Proportional Reasoning

Standard 8-2:The student will demonstrate through the mathematical processes an understanding of operations with integers, the effects of multiplying and dividing with rational numbers, the comparative magnitude of rational and irrational numbers, the approximation of cube and square roots, and the application of proportional reasoning.

Indicators:

8-2.1Apply an algorithm to add, subtract, multiply, and divide integers.

8-2.2Understand the effect of multiplying and dividing a rational number by another rational number.

8-2.3Represent the approximate location of irrational numbers on a number line.

8-2.4Compare rational and irrational numbers by usingthe symbols ≤, ≥, <, >, and =.

8-2.5Apply the concept of absolute value.

8-2.6Apply strategies and procedures to approximate between two whole numbers the square roots or cube roots of numbers less than 1,000.

8-2.7Apply ratios, rates, and proportions.

Essential Questions:

How can you model the four basic operations with integers? (8-2.1)
What happens to a rational number when multiplied or divided by a rational number? (8-2.2)
How can you justify the approximate location of an irrational number on a number line? (8-2.3)
How do you differentiate between rational and irrational numbers? (8-2.4)
Why is the absolute value of a number and its opposite the same? (8-2.5)
What is the difference between cubing and squaring a number? (8-2.6)
How can we use ratios, rates and proportions to solve real-world problems? (8-2.7)

Help Page for Standard 8-2

Notes:

Assessment examples can be accessed at
Module 1-1 (8-2.6, 8-2.3, 8-2.4, 8-2.5)
Module 1-2 (8-2.1, 8-2.2, 8-2.7)
Formative Assessment is embedded within the lesson through questioning and observation; however, other formative assessment strategies should be employed.
Assessment Examples:
Chapter Review and Test Prep
Chapter Tests
MAP Testing
Odyssey
Questioning Strategies
Exit tickets
Journaling/Written Assessment
Projects
Pair Shares
Textbook Correlations
8-2.1 Lesson 1-4 (p. 18)
Lesson 1-5 (p. 22)
Lesson 1-6 (p. 26)
8-2.2Lesson 2-4 (p. 76)
Lesson 2-3 (p.72)
Lesson 2-5 (p. 80)
Lesson 2-6 (p. 85)
8-2.3 Lesson 4-7 (p. 191)
Lesson 2-1 (p. 64)
8-2.4 Lesson 2-2 (p. 68)
WT (p. 112-114)
8-2.5 Lesson 1-3 (p. 14)
8-2.6 Lesson 4-6 (p. 186, 830)
WT (p. 110-111)
Lesson 4-1 (p. 162)
Lesson 4-2 (p. 166)
Lesson 4-3 (p. 170)
Textbook Correlations (continued)
8-2.7 Lesson 5-1 (p. 216)
Lesson 5-2 (p. 220)
Lesson 5-4 (p. 229)
Key Concepts (Vocabulary)
integers
rational numbers
irrational numbers
absolute value / square roots
cube roots
ratios
rates
proportions
Literature

Gulliver’s Travels by Swift
Holes by Louis Sachar
The Principal’s New Clothes by Calmenson
Saturday Night at the Dinosaur Stomp by Sheilds
Tuck Everlasting by Natalie Babbitt
Cut Down to Size at High Noon: A Math Adventure by Sundby
Best of Times by Tang
Uno’s Garden by Base
Math Doesn’t Suck: How to Survive Middle School Math Without Losing Your Mind or Breaking a Nail by McKellar
The 512 Ants on Sullivan Street by Carol A. Losi
Jim and the Beanstalk by Raymond Briggs
One Hundred Hungry Ants by Elinor J. Pinczes
Roll of Thunder, Hear My Cry by Mildred D. Taylor

Technology
One Stop Planner Videos
Supporting Content Web Sites

Lessons, Activities, and Related Web Links
(National Library of Virtual Manipulatives)
(ETV Streamline and more!)

Supporting Content Web Sites (continued)

Interactive Lessons and Activities based on our 2007 SC Mathematics Standards and Indicators

Number Structure and Relationships - Rational and Irrational Numbers

Indicators

8-2.3Represent the approximate location of irrational numbers on a number line.

8-2.4Compare rational and irrational numbers by usingthe symbols ≤, ≥, <, >, and =.

8-2.5Apply the concept of absolute value.

8-2.6 Apply strategies and procedures to approximate between two whole

numbers the square roots or cube roots of numbers less than 1,000.

Eighth grade is the first time students are introduced to the concept of irrational numbers. Students will continue to build on prior knowledge by now representing the approximate location of irrational numbers on a number line. Since an irrational number can be approximated as a decimal, it can also be approximately placed on a number line. Students are fine tuning their estimation skills and rounding abilities when they are choosing the location for an irrational number on a number line. Students should be aware that every number on the number line is either rational or irrational. It is also important to note that NAEP uses pi as a “benchmark” irrational number.

In seventh grade, work with squares and roots involved perfect squares (integers). Therefore, emphasis should be placed on comparing rational and irrational numbers by using the symbols ≤, ≥, <, >,and =. Discussion should include why numbers are rational vs. irrational. A rational number can be represented as a ratio of two integers where the denominator does not equal zero and an irrational number can not be written as a ratio of two integers. Irrational numbers when written as decimals do not terminate or have a repeating pattern the way rational numbers do.

Seventh grade students developed an understanding for the meaning of absolute value. Beginning in eighth grade, students will apply the concept of absolute value. With an understanding of what absolute value means, students can work examples that involve finding the absolute value. Examples could include the difference between below and above sea level, freezing temperatures which are below and above zero, being in the red as a deficit balance and then in the black for having a positive bank balance, deposits and withdrawals, cars that are going in different directions, yards gained or lost in a football game, etc.

In seventh grade, students learned about the relationship between squaring and finding square roots of perfect square numbers. Now students will apply strategies and procedures to approximate between two whole numbers the square roots or cube roots of numbers less than 1,000. Students will need to develop conceptual understanding and be expected to move to fluency. Cube roots will be a new topic for students but one that understanding can build from prior knowledge of square roots. As examples: the square root of 50 is between 7 and 8, the cube root of 25 is between 2 and 3. Once students learn how to find the square root of all numbers, they will be able to find missing sides of a right triangle using the Pythagorean Theorem. It will also be easier for them to use the quadratic formula to solve quadratic equations in Algebra 1.

Operations and Proportional Reasoning

Indicators

8-2.1Apply an algorithm to add, subtract, multiply, and divide integers.

8-2.2Understand the effect of multiplying and dividing a rational number by another rational number.

8-2.7 Apply ratios, rates, and proportions.

Eighth grade expands the application of computational skills to all operations involving integers. Students in seventh grade generated strategies to add, subtract, multiply, and divide integers. As a result of sharing those generated strategies, students developed a conceptual understanding of integer operations. In other words, student work with integers was limited to concrete and pictorial models. Therefore, it is beneficial to discuss strategies students developed with pictorial representation before moving into the algorithm. The emphasis for eighth grade is to apply an algorithm to add, subtract, multiply, and divide. As a result, by the end of eighth grade student should exhibit fluency when solving a wide range of addition, subtraction, multiplication, and division problems involving integers.

Eighth grade students should be able to use all operations to solve a variety of problems involving integers and explain the meaning and effects of those operations when working with integers. Operations should also be introduced in context so that students develop an understanding of the algorithm instead of being taught arbitrary procedures. Teachers should refer to the additive and multiplicative inverses as instructing. Teachers will need to provide a solid understanding of addition prior to subtraction and multiplication prior to division.

Students will extend learning of multiplication and division by using rational numbers. Specifically, teachers will need to provide experiences for students to see the effect of multiplying and dividing a rational number by zero, greater than one, and a number between zero and one. Students have difficulty in understanding that division by zero is impossible. Discussion should focus on the resulting effect on the product and quotient when the operation is performed.

In seventh grade, students applied ratios, rates, and proportions to discounts, taxes, tips, interest, unit costs, and similar shapes. Eighth grade students will continue to apply ratios, rates, and proportions in problem solving situations.

When using cross-multiplying as a method for solving problems, teachers should provide students with meaningful experiences to develop a deep understanding. Cross multiplication is a powerful technique but needs to be taught with understanding. One way to develop cross multiplying is to create equivalent ratios by multiplying each term by 1:

2 = a3 x 2 = 7 x a3 x 2 = 7 x a

7 33 x 7 7 x 3

Grade 8

Strand - Algebra

Big Ideas: Equations, Inequalities, Slope, Linear Functions, and Properties

Standard 8-3:The student will demonstrate through the mathematical processes an understanding of equations, inequalities, and linear functions.

Indicators:

8-3.1Translate among verbal, graphic, tabular, and algebraic representations of linear functions.

8-3.2Represent algebraic relationships with equations and inequalities.

8-3.3Use commutative, associative, and distributive properties to examine the equivalence of a variety of algebraic expressions.

8-3.4Apply procedures to solve multi-step equations.

8-3.5Classify relationships between two variables in graphs, tables, and/or equations as either linear or nonlinear.

8-3.6Identify the coordinates of the x- and y-intercepts of a linear equation from a graph, equation, and/or table.

8-3.7Identify the slope of a linear equation from a graph, equation, and/or table.

Essential Questions:

How can linear functions be transferred from one representation to another? (8-3.1)
Are there things or relationships for which equations cannot be used? (8-3.2)
Why did mathematicians insist upon asserting certain properties in the use of algebraic equations? (8-3.3)
How do you determine which steps are necessary to solve multi-step equations? (8-3.4)
How can algebraic solutions be visualized on graphs? (8-3.5)
How do you use graphs, tables, and equations to identify intercepts and slope? (8-3.6, 8-3.7)

Help Page for Standard 8-3

Notes:

Assessment examples can be accessed at
Module 2-1 (8-3.2, 8-3.3)
Module 2-2 (8-3.1, 8-3.4, 8-3.6, 8-3.7)
Module 2-3 (8-3.5)
Formative Assessment is embedded within the lesson through questioning and observation; however, other formative assessment strategies should be employed.
Assessment Examples:
Chapter Review and Test Prep
Chapter Tests
MAP Testing
Odyssey
Questioning Strategies
Exit tickets
Journaling/Written Assessment
Projects
Pair Shares
Textbook Correlations
8-3.1 Lesson 1-2 (p. 10)
Lesson 13-4 (p. 700)
8-3.2 N.A. (reference SC default curriculum Module 2-1)
8-3.3 WT (p. 3-6)
8-3.4 Lesson 11-2 (p. 588)
8-3.5 Lesson 12-1 (p. 628)
8-3.6 Lesson 12-3 (p. 638)
8-3.7 Lesson 12-2 (p. 633)
Lesson 12-3 (p. 638)
Key Concepts (Vocabulary)
multi-step equations
inequalities
linear relationships
non-linear relationships
Key Concepts (Vocabulary) (continued)
functions
inequalities
equations / commutative
associative
distributive
slope
coordinates
x-intercept
y-intercept
variables
Literature

A Gebra Named Al by Isdell
Kiss My Math: Showing Pre-Algebra Who’s Boss by McKellar
Secrets, Lies, and Algebra by Lichtman
The Fly on the Ceiling by Glass

Technology
One Stop Planner Video
Supporting Content Web Sites
Suggested Streamline Videos

MathsMansion 36: The X and Y Files(plotting points in a coordinate plane)
The Adventurous World of Alg: Program 3: Linear Equations(graphing linear equations, finding slope)
The Pumped up World of Pre-Algebra, Program 4: Basic Algebra
The Pumped up World of Pre-Algebra, Program 5: Algebra Equations
Linear Equations: y = mx + b
Operations Algebra: Introduction to Linear Relations(identify linear relations as equations, tables, and graphs)
Operations Algebra: Linear Equations Ax + By = C
Operations Algebra: Solving Equations with Variables on Both Sides of the Equation
Using Tables to Graph Equations

Cross Curricular Opportunities
Health and Fitness
Science
Social Studies
Art
Career Connections
Ecologist
Marketing
Business

Eighth Grade---Support Document

Algebra

Standard 8-3: The student will demonstrate through the mathematical processes

an understanding of equations, inequalities, and linear functions.

The indicators for this standard are grouped by the following major concepts:

Patterns, Relationships, and Functions
Representations, Properties, and Proportional Reasoning
Solve Mathematical Situations

The indicators that support each of those major concepts and an explanation of the essential learning for each major concept follows.

Patterns, Relationships, and Functions

Indicator

8-3.5 Classify relationships between two variables in graphs, tables, and/or equations as either linear or nonlinear.

In 7th grade, students analyzed tables and graphs to describe the rate of change among quantities and to determine if the rate of change is constant or not. They learned that if the rate of change is constant, this is also called the slope, and that when graphed the result is a straight line. While the introduction of slope takes place in 7th grade, it is in 8th grade that students gain the ability to identify and classify relationships between two variables in graphs, tables, and/or equations as linear or nonlinear and to compare and contrast the properties of each. Students in the 8th grade should have the opportunity to discover that a linear relationship will have a constant rate of change. In regards to equations, students must learn that a linear equation has no operations other than addition, subtraction, and multiplication of a variable by a constant, that the variables cannot be multiplied or appear in the denominator, and that a linear equation cannot contain variables with exponents other than one. If a relationship does not fit these criteria, students should recognize the relationship as nonlinear.