MATHEMATICS – GRADE 8 Idaho State

IDAHO MATHEMATICS PERFORMANCE STANDARDS GRADE 8

(DRAFT I)

NOTE: The samples associated with the content standards are meant to illustrate meaning and to represent possible areas of applications. They are not intended to be an exhaustive list, but are samples of applications that would demonstrate learning.

I. BASIC ARITHMETIC, ESTIMATION, AND ACCURATE COMPUTATIONS Rationale: An understanding of numbers and how they are used is necessary in the everyday world. Computational skills and procedures should be developed in context so the learner perceives them as tools for solving problems.
Standard – The student will:	Content Knowledge and Skills:	Samples of Applications:
01. Understand and use numbers.	a. Understand and use fractions, decimals, percents, and ratios to solve problems.	i. Interchange fractions, decimals, or percents, to determine pricing markup or markdown. ii. Use fractions to increase or decrease recipes. iii. Use percentages to compute taxes and commissions. iv. Use decimals to compute interest problems. v. Use ratios to compute sports averages (batting averages).
	b. Understand and use real numbers, both rational and irrational.	i. Explain that irrational numbers neither terminate nor repeat when written in decimal form. ii. Calculate the square root of a number to the thousandths.
	c. Show a sense of magnitudes and relative magnitudes of numbers using scientific notation, fractions, decimals and exponential numbers.	i. Express distances in our solar system using scientific notation. ii. Order a list of fractions, decimals, and exponential numbers.
	d. Develop and apply number theory concepts; e.g., primes, factors, and multiples in content problem situations.	i. Identify prime factorization, greatest common factors and least common multiples.
	e. Understand the position of real numbers on a number line.	i. Locate a given set of real numbers on a number line. ii. Accurately identify positive and negative integers on a thermometer.
02. Perform computations accurately.	Consistently and accurately add, subtract, multiply, and divide rational numbers.	i. Plan a road trip. Record and calculate gas mileage, food and lodging expenses.
	a. Understand and use both exponents.	i. Solve problems using exponential growth, i.e., Sally tells three friends a secret. The next day those three friends tell three other friends. If this pattern continues for five days, how many friends will hear the secret?
	b. Evaluate numerical expressions using the order of operations.	i. Find the surface area of a cylinder.
03. Estimate and judge reasonableness of results.	a. Estimate using decimals and fractions.	i. Estimate the total cost of a back-to-school shopping spree.
	b. Recognize when estimation is appropriate and understand the usefulness of an estimate as distinct from an exact answer.	i. Estimate the population of people at the mall on any given date. Ii Explain why or when exact answers might be necessary.
	c. Determine whether a given estimate is an overestimate or underestimate.	i. Estimate driving time to a given destination and compare to actual time (given particular miles per hour).

II. MATHEMATICAL REASONING AND PROBLEM SOLVING Rationale: These processes are essential to all mathematics and must be incorporated in all other mathematics standards.
Standard – The student will:	Content Knowledge and Skills:	Samples of Applications:
01. Understand and use a variety of problem-solving skills.	a. Use a variety of strategies including common mathematical formulas to compute problems drawn from real life situations.	i. Determine amount of interest earned in a savings account. Ii Compute amount of paint needed for a bedroom. iii. Given $500 to plan a year-end class party using several committees develop a budget for the party.
	b. Recognize pertinent information for problem solving.	i. Given a problem, identify essential information.
02. Use reasoning skills to recognize problems and express them mathematically.	a. Apply solutions and strategies to new problem situations.	i. After finding the area of square, find the surface area of a cube.
	b. Formulate conjectures and justify (short of formal proof) why they must be or seem to be true.	i. Debate the following: If people like cheese and mice like cheese, then people are mice. ii "On a crisp autumn day, the first four students to class are wearing shorts." What will be the possible attire of the rest of the students in the class? Explain.
03. Apply appropriate technology and models to find solutions to problems.	a. Understand the purpose and capabilities of appropriate technology.	i. Use technology to find the square root of 5,281.
	b. Use computer applications to display and manipulate data.	i. Display and compare the frequency of various M&M colors using computer applications.
04. Communicate results using appropriate terminology and methods.	a. Select appropriate means to communicate mathematical information.	i. Write an explanation on how to solve a problem. Ii Display results of a problem in graphic form.

III. CONCEPTS AND PRINCIPLES OF MEASUREMENT Rationale: The first step in scientific investigation is understanding the measurable attributes of objects.
Standard – The student will:	Content Knowledge and Skills:	Samples of Applications:
01. Understand and use United States. Customary and metric measurements.	a. Select and use appropriate units and tools to make measurements using both systems.	i. Select and use the appropriate tool to measure the length of a school building, the length of a classroom, the length of a thumbtack and the length of a student’s shoe. Ii Measure the interior angles of a triangle.
	b. Recognize the differences between measures of length, area, and volume (capacity) in both systems.	i. Model length, area, and volume using various manipulatives, i.e., string, tiles, and cubes.
	c. Solve problems involving length, perimeter, area, volume (capacity), angle measure, weight, mass, and temperature.	i. Design and determine how much lumber would be needed for an average size dog house. Ii Design a laser alarm system using numerous mirrors as reflection points.
02. Apply concepts of rates and other derived or indirect measurements.	a. Use rates to make indirect measurements.	i. Use formulas and appropriate procedures for solving measurement problems, i.e., distance equals rate * time. Ii Given the weight of 1-gallon water, calculate the weight of water in the swimming pool.
03. Apply the concepts of ratios and proportions.	a. Understand and use proportions, ratios, and scales.	i. Use proportions to find the height of a flagpole. Ii Use map scale to determine distances. Iii Enlarge a cartoon to a given scale.
04. Apply dimensional analysis.	a. Understand units and their relationship to one another and to real world applications.	i. Determine the reading hours of an entire class if everyone reads twenty minutes.

IV. CONCEPTS AND LANGUAGE OF ALGEBRA Rationale: Algebra is the language of mathematics and science. Through the use of variables and operations, algebra allows students to form abstract models from contextual information.
Standard – The student will:	Content Knowledge and Skills:	Samples of Applications:
01. Use algebraic symbolism as a tool to represent mathematical relationships.	a. Understand and use variables in expressions, equations, and inequalities.	i. If B represents the number of boys in the class, and G represents the number of girls in the class, write an equation and solve it in the number of students in the classroom.
	b. Translate word statements and story problems into algebraic expressions and equations.	i. Given that the price of gas is changing each week, write an algebraic expression to represent the cost of purchasing 10 gallons of gas.
02. Evaluate algebraic expressions.	a. Understand and use the following properties in evaluating algebraic expressions: distributive, associative, commutative, inverse, identity, and substitution.	i. Evaluate expressions such as 2x + y when given values for x and y.
	a. Understand and use the order of operations in evaluating algebraic expressions.	i. Explain why order of operations is important when evaluating expression such as: 3 + 2 * 8.
	b. Simplify algebraic expressions with like terms.	i. Simplify expressions such as: 3a + 4b –15a + 6b –7.
03. Solve algebraic equations and inequalities.	a. Solve one and two step equations and inequalities using inverse operations.	i. Solve equation such as: 12 x 15 = 31.
	b. Use graphical representations to show linear equations.	i. Use technology to create a graph of linear relations.

V. CONCEPTS AND PRINCIPLES OF GEOMETRY Rationale: The study of geometry helps students represent and make sense of the world by discovering relationships and developing spatial sense.
Standard – The student will:	Content Knowledge and Skills:	Samples of Applications:
01. Apply concepts of size, shape, and spatial relationships.	a. Understand, model, and apply the effects of reflections, translations, and rotations on various shapes.	i. Create a tessellation using a combination of reflection, translation or rotation showing movement on a plane.
	b. Understand and apply fundamental concepts, properties and relationships among points, lines, planes, angles, and shapes.	i. Compare measurements of opposite and alternate angles in a parallelogram.
	c. Apply formulas for perimeter, circumference and area to polygons, and circles.	i. Calculate the area of a center circle of a basketball court.
	d. Understand the concept of surface area and volume (capacity).	i. Construct a cylinder and calculate its surface area and volume (capacity).
02. Apply the geometry of right triangles.	a. Investigate right triangle geometry using the Pythagorean Theorem and 30-60 and 45-45 right triangles.	i. Investigate the relationships among the three sides of a right triangle of a Pythagorean triple. Ii Determine the length of a handicap access ramp given the height and length of the incline.
03. Apply graphing in two dimensions.	a. Represent geometric figures on a coordinate grid.	i. Construct a geometric figure given the following set of ordered pairs: (1,1), (1, -3), (-2, -3), (-2, 1).

VI. DATA ANALYSIS, PROBABILITY AND STATISTICS Rationale: With society’s expanding use of data for prediction and decision making, it is important that students develop an understanding of the concepts and processes used in analyzing data
Standard – The student will:	Content Knowledge and Skills:	Samples of Applications:
01. Understand data analysis.	a. Analyze and interpret tables, charts, and graphs; e.g., scatter plots, line graphs, bar graphs, and pie charts.	i. Analyze and interpret bar graphs and pie charts in magazines or newspapers. Ii Given a scatter plot, choose a line of best fit.
02. Collect, organize, and display data.	a. Collect and organize data, and display the data in tables, charts, and graphs; e.g., scatter plots, line graphs, bar graphs, and pie charts.	i. Collect and display data by surveying the actual number of chocolate chips in cookies. Ii Survey and graphically display music preference categories.
03. Apply simple statistical measurements.	a. Choose and calculate the appropriate measure of central tendency mean, median, and mode.	i. Compute an individual mean grade in a particular course. Ii Find the median student height in a classroom. Iii Find the most common shoe size in your class.
	b. Understand the significance of range, frequency, and informal distribution.	i. Calculate range and distribution of professional basketball salaries. Ii Create a frequency distribution chart of professional basketball salaries.
04. Understand basic concepts of probability.	a. Recognize equally likely outcomes.	i. Document frequency of heads and tails when flipping a coin 100 times.
	b. Model situations of probability using simulations.	i. Find the probability of drawing a jack of hearts in a standard deck of cards. Ii Use a spinner to represent the number of points scored by a 75% free throw shooter in a Basketball game.
05. Make predictions or decisions based on data.	a. Make predictions based on experimental and theoretical probabilities.	i. Find the theoretical and experimental probability of throwing a pair of six sided number cubes and getting a sum of 7.
	b. Conduct and interpret results of statistical experiments using tables, charts, or graphs.	i. Graphically display the results of 40 spins of a four-quadrant spinner, and explain the results.

VII. FUNCTIONS AND MATHEMATICAL MODELS Rationale: One of the central themes of mathematics is the study of patterns, relationships, and functions. Exploring patterns helps students develop mathematical power.
Standard – The student will:	Content Knowledge and Skills:	Samples of Applications:
01. Understand the concept of functions.	a. Extend patterns and identify a rule that generates the pattern.	i. Extend the first five numbers in the Fibonacci sequence and explain the rule. Ii Explain how the amount of sales tax depends on the cost of the item.
	b. Understand and use functional relationships to explain how a change in one quantity results in a change in another.	i. Find the number of pencils needed for a classroom if each student must have three pencils. Explain how the number of pencils needed will change as the number of students change.
02. Represent equations, inequalities, and functions in a variety of formats.	a. Represent a set of data in a table, as a graph, and as a mathematical relationship.	i. Collect data on the temperature of hot water as it cools over time and organize and display the results in several forms, including tables and graphs.
03. Apply functions to a variety of problems.	a. Use patterns and functions to represent and solve problems.	i. Illustrate on a line graph distance traveled at a constant speed of 55 miles per hour.

Performance Standards - Grade 8 Mathematics

Advanced

Eighth-grade students performing at the advanced level should be able to consistently demonstrate their ability to go beyond the obvious when applying mathematical processes. They effectively use multiple strategies and extend concepts to new situations. They explore hypothetical questions and articulate valid arguments.

Advanced eighth graders demonstrate their ability to go beyond the obvious when applying number, operations, patterns, relations and functions. They consistently apply algebraic concepts and processes to solve real and theoretical problems. They can go beyond the obvious understanding of and ability to use data analysis, probability, statistics, shape, geometry and measurement while recognizing the interconnections within and outside mathematics.

Proficient

Eighth graders performing at the proficient level consistently apply mathematical processes correctly using a variety of problem-solving strategies. They demonstrate valid reasoning. Estimation skills are appropriate and consistent. They consistently apply mathematics in a variety of contexts. Using appropriate mathematical terms, proficient eighth-grade students formulate logical arguments. They consistently select and use appropriate technology to enhance mathematical understanding.

Proficient eighth graders consistently demonstrate their ability to accurately use number, operations, patterns, relations and functions. They commonly use algebraic concepts and processes to represent and solve mathematical problems. They consistently demonstrate their ability to accurately use data analysis, probability, statistics, shape, geometry and measurement.

Satisfactory

Eighth-grade students performing at the satisfactory level commonly apply mathematical processes correctly using a number of problem-solving strategies. They demonstrate satisfactory reasoning. Estimation skills are appropriately applied. They commonly apply mathematics in a variety of contexts. Using appropriate mathematical terms, satisfactory eighth-grade students formulate logical arguments. They commonly select and use appropriate technology to enhance mathematical understanding.

Satisfactory eighth graders commonly demonstrate their ability to accurately use number, operations, patterns, relations and functions. They often use algebraic concepts and processes to represent and solve mathematical problems. They frequently demonstrate their ability to accurately use data analysis, probability, statistics, shape, geometry and measurement.

Developing

Eighth-grade students performing at the developing level sporadically apply mathematical processes using appropriate problem-solving strategies. Demonstrate low-level reasoning skills. Their ability to estimate is inconsistent. Their communication of mathematical ideas is approaching completeness, but may be inaccurate. Some connections are made, but not generalized. They begin to apply mathematics to recognize its relevance, both in and out of school. They often use technology appropriately.

Eighth graders at this level often demonstrate an understanding of and ability to use number, operations, patterns, relations and functions. They occasionally use algebraic concepts and processes to represent and solve mathematical problems. They often, but not always, demonstrate an understanding of data analysis, probability, statistics, shape, geometry and measurements.

Minimal

Eighth-grade students performing at the minimal level rarely select and apply mathematical processes using appropriate problem solving strategies. Demonstrate minimal reasoning. They use inappropriate estimation strategies; their communication of mathematical ideas is limited, incomplete and unclear. Only immediate, concrete connections are made. They lack the ability to apply mathematics or recognize its relevance, both in and out of school. They seldom use technology appropriately or with understanding.

Novice eighth graders sometimes demonstrate an understanding of and an ability to use number, operations, patterns, relations and functions. They may use minimal algebraic concepts and processes to represent and solve mathematical problems. They seldom demonstrate understanding of data analysis, probability and statistics. They sometimes demonstrate an understanding of shape, geometry and measurement.