CCGPS Frameworks Student Edition. Mathematics. 8 th Grade Unit 2: Exponents and Equations

Transcription

1 CCGPS Frameworks Student Edition Mathematics 8 th Grade Unit 2: Exponents and Equations These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. The contents of this guide were developed under a grant from the U. S. Department of Education. However, those contents do not necessarily represent the policy of the U. S. Department of Education, and you should not assume endorsement by the Federal Government.

2 TABLE OF CONTENTS Unit 2 EXPONENTS and EQUATIONS Overview...3 Key Standards & Related Standards...4 Enduring Understandings...5 Concepts & Skills to Maintain...6 Selected Terms and Symbols...6 Formative Assessment Lesson (FAL) Overview...8 Tasks...9 Rational or Irrational Reasoning?...9 FAL: Repeating Decimals...10 A Few Folds Alien Attack...13 Nesting Dolls...16 Exponential Exponents...17 Exploring Powers of SCT: Ponzi Pyramid Scheme...21 FAL: Estimating Length Using Scientific Notation...23 E. coli...25 SCT: Giantburgers...26 SCT: 100 People...28 FAL: Solving Linear Equations in One Variable...30 Writing for a Math Website...32 July 2013 Page 2 of 33

3 OVERVIEW In this unit student will: distinguish between rational and irrational numbers and show the relationship between the subsets of the real number system; recognize that every rational number has a decimal representation that either terminates or repeats; recognize that irrational numbers must have decimal representations that neither terminate nor repeat; understand that the value of a square root can be approximated between integers and that non perfect square roots are irrational; locate rational and irrational numbers on a number line diagram; use the properties of exponents to extend the meaning beyond counting-number exponents; recognize perfect squares and cubes, understanding that non-perfect squares and nonperfect cubes are irrational; recognize that squaring a number and taking the square root of a number are inverse operations; likewise, cubing a number and taking the cube root are inverse operations; express numbers in scientific notation; compare numbers, where one is given in scientific notation and the other is given in standard notation; compare and interpret scientific notation quantities in the context of the situation; use laws of exponents to add, subtract, multiply and divide numbers written in scientific notation; solve one-variable equations with the variables being on both sides of the equals sign, including equations with rational numbers, the distributive property, and combining like terms; and analyze and represent contextual situations with equations, identify whether there is one, none, or many solutions, and then solve to prove conjectures about the solutions. Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as estimation, mental computation, and basic computation facts should be addressed on an ongoing basis. Ideas related to the eight practice standards should be addressed constantly as well. To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under Evidence of Learning be reviewed early in the planning process. A variety of resources should be utilized to supplement this unit. This unit provides much needed content information, but excellent learning activities as well. The tasks in this unit illustrate the types of learning activities that should be utilized from a variety of sources. July 2013 Page 3 of 33

4 STANDARDS ADDRESSED IN THIS UNIT Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics. KEY STANDARDS Work with radicals and integer exponents. MCC8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, ( 5) = 3 ( 3) 1 = = MCC8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 =p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. MCC8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as and the population of the world as , and determine that the world population is more than 20 times larger. MCC8.EE.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. MCC8.EE.7 Solve linear equations in one variable. a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = b, or a = a results (where a and b are different numbers). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. MCC8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. July 2013 Page 4 of 33

5 MCC8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π 2 ). For example, by truncating the decimal expansion of 2 (square root of 2), show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. STANDARDS FOR MATHEMATICAL PRACTICE Refer to the Comprehensive Course Overview for more detailed information about the Standards for Mathematical Practice. 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ENDURING UNDERSTANDINGS Square roots can be rational or irrational. An irrational number is a real number that cannot be written as a ratio of two integers. Every number has a decimal expansion, for rational numbers it repeats eventually, and can be converted into a rational number. All real numbers can be plotted on a number line. Rational approximations of irrational numbers can be used to compare the size or irrational numbers, locate them approximately on a number line, and estimate the value of expressions. 2 is irrational. Exponents are useful for representing very large or very small numbers. Properties of integer exponents can be use to generate equivalent numerical expressions. Scientific notation can be used to estimate very large or very small quantities and to compare quantities. July 2013 Page 5 of 33

6 Linear equations in one variable can have one solution, infinitely many solutions, or no solutions. CONCEPTS/SKILLS TO MAINTAIN It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas. computation with whole numbers and decimals, including application of order of operations solving equations plotting points in four quadrant coordinate plane understanding of independent and dependent variables characteristics of a proportional relationship SELECTED TERMS AND SYMBOLS The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them. The definitions below are for teacher reference only and are not to be memorized by the students. Students should explore these concepts using models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers. The websites below are interactive and include a math glossary suitable for middle school students. Note Different sources use different definitions. Please preview any website for alignment to the definitions given in the frameworks. The definitions below are from the Common Core State Standards Mathematics Glossary and/or the Common Core GPS Mathematics Glossary when available. Visit or to see additional definitions and specific examples of many terms and symbols used in grade 8 mathematics. Addition Property of Equality: For real numbers a, b, and c, if a = b, then a + c = b + c. In other words, adding the same number to each side of an equation produces an equivalent equation. July 2013 Page 6 of 33

7 Additive Inverses: Two numbers whose sum is 0 are additive inverses of one another. Example: 3/4 and 3/4 are additive inverses of one another because 3/4 + ( 3/4) = ( 3/4) + 3/4 = 0. Algebraic Expression: A mathematical phrase involving at least one variable. Expressions can contain numbers and operation symbols. Addition Property of Equality: For real numbers a, b, and c, if a = b, then a + c = b + c. In other words, adding the same number to each side of an equation produces an equivalent equation. Cube Root: If the cube root of a number b is a (i.e., 3 b = a), then a 3 = b. Decimal Expansion: The decimal expansion of a number is its representation in base 10 (i.e., the decimal system). For example, the decimal expansion of 25 2 is 625, of π is , and of 1/9 is Equation: A mathematical sentence that contains an equals sign. Evaluate an Algebraic Expression: To perform operations to obtain a single number or value. Exponent: The number of times a base is used as a factor of repeated multiplication. Exponential Notation: See Scientific Notation below. Inverse Operation: Pairs of operations that undo each other, for example, addition and subtraction are inverse operations and multiplication and division are inverse operations. Irrational: A real number whose decimal form is non-terminating and non-repeating that cannot be written as the ratio of two integers. Like Terms: Monomials that have the same variable raised to the same power. Only the coefficients of like terms can be different. Linear Equation in One Variable: An equation that can be written in the form ax + b = c where a, b, and c are real numbers and a 0 Multiplication Property of Equality: For real numbers a, b, and c (c 0), if a = b, then ac = bc. In other words, multiplying both sides of an equation by the same number produces an equivalent expression. July 2013 Page 7 of 33

8 Multiplicative Inverses: Two numbers whose product is 1 are multiplicative inverses of one another. Example: 3/4 and 4/3 are multiplicative inverses of one another because 3/4 4/3 = 4/3 3/4 = 1. Perfect Square: A number that has a rational number as its square root. Radical: A symbol that is used to indicate square roots. Rational: A number that can be written as the ratio of two integers with a nonzero denominator. Scientific Notation: A representation of real numbers as the product of a number between 1 and 10 and a power of 10, used primarily for very large or very small numbers. Significant Digits: A way of describing how precisely a number is written. Solution: the value or values of a variable that make an equation a true statement Solve: Identify the value that when substituted for the variable makes the equation a true statement. Square Root: One of two equal factors of a nonnegative number. For example, 5 is a square root of 25 because 5 5 = 25. Another square root of 25 is -5 because (-5) (-5) = 25. The +5 is called the principle square root of 25 and is always assumed when the radical symbol is used. Variable: A letter or symbol used to represent a number. FORMATIVE ASSESSMENT LESSONS (FALs) Formative Assessment Lessons are intended to support teachers in formative assessment. They reveal and develop students understanding of key mathematical ideas and applications. These lessons enable teachers and students to monitor in more detail their progress towards the targets of the standards. They assess students understanding of important concepts and problem solving performance, and help teachers and their students to work effectively together to move each student s mathematical reasoning forward. More information on types of Formative Assessment Lessons may be found in the Comprehensive Course Guide. July 2013 Page 8 of 33

9 SE TASK: Rational or Irrational Reasoning? Analyze Robin s reasoning in her answer to the test question about rational or irrational numbers. Does she have a deep understanding of rational and irrational numbers? Does her reasoning make sense? If not, what misconceptions does she have about this topic? Create a study guide with explanations, examples, and graphics to help clear up any misconceptions students might have over these topics. 3 rd Block Robin Radical 1. Is 130 rational or irrational? Where would 130 be located on a number line? Explain your reasoning. 130 is a rational number because 130 is even. All rational numbers are even and irrational numbers are odd when the numbers are under the square root sign. The square root sign is the opposite of squaring a number. Squaring a number is the same as raising it to the 2 nd power. So, to find the value of a number under the square root sign, you divide it by 2. So, is 65. The answer is 65. July 2013 Page 9 of 33

10 Formative Assessment Lesson: Repeating Decimals Source: Formative Assessment Lesson Materials from Mathematics Assessment Project In this FAL, students will translate between decimal and fraction notation (particularly when the decimals are repeating), create and solve simple linear equations to find the fractional equivalent of a repeating decimal, and understand the effect of multiplying a decimal by a power of 10. STANDARDS ADDRESSED IN THIS TASK: MCC8.EE.7 Solve linear equations in one variable. a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = b, or a = a results (where a and b are different numbers). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. MCC8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. MCC8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π 2 ). For example, by truncating the decimal expansion of 2 (square root of 2), show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. STANDARDS FOR MATHEMATICAL PRACTICE: This task uses all of the practices with emphasis on: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 6. Attend to precision. July 2013 Page 10 of 33

11 TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION: Tasks and lessons from the Mathematics Assessment Project are specifically designed to help teachers effectively formatively assess their students. The way the tasks and lessons are designed gives the teacher a clear understanding of what the students are able to do and not do. Within the lesson, teachers will find suggestions and question prompts that will help guide students towards understanding. For more information access the MAP website: The task, Repeating Decimals, is a Formative Assessment Lesson (FAL) that can be found at the website: The FAL document provides a clear lesson design, from the opening of the lesson to the closing of the lesson. The PDF version of the task can be found at the link below: July 2013 Page 11 of 33

12 Part 1: SE TASK: A Few Folds Repeatedly fold one piece of paper in half, recording the number of folds and the resulting number of layers of paper. Assuming that you could continue the pattern, how many layers of paper would there be for 10 folds, 100 folds, n folds? How do you know? Part 2: Explain the process that is used to generate the following patterns? Cup, pint, quart, half gallon, and gallon, and so on. Penny, dime, dollar, ten dollars, one hundred dollars, and so on. July 2013 Page 12 of 33

13 Alien Attack! SE TASK: Aliens from the Outer Space Galactic Task Force have been watching the recent gains in mathematical understanding developing in human brains from Planet Earth. They have become alarmed that Planet Earth might soon develop the capabilities to discover that life exists on other planets with this increased mathematical knowledge. To slow the progress, they have organized an attack on the World Wide Web and all other forms of math textbooks. All words have been eradicated from the mathematics examples! Humans will now be forced to study the patterns of numbers and previously worked examples to rediscover the properties of mathematics! They are confident that humans will not persevere in the challenge of making sense of problems, reasoning abstractly and quantitatively, constructing viable arguments, looking for and making use of structure, and using repeated reasoning to recreate the language of mathematics. They have already declared MISSION ACCOMPLISHED! The next standard in your mathematics class requires knowledge of the properties of integer exponents. Can your team recreate and name the properties by using the Standards for Mathematical Practice to examine the examples that remain? Problem continues on next page. July 2013 Page 13 of 33

14 Problem continued from previous page = (7 7 7) (7 7) = = = (5 5 5) (5 5 5) = = = (3) ( ) = 3 5 (2 2 ) 3 = (2 2 ) (2 2 ) (2 2 ) = (2 2) (2 2) (2 2) = = 2 6 (6 4 ) 2 = (6 4 ) (6 4 ) = ( ) ( ) = = = (3 2) 5 = = (4 5) 3 = 20 3 (2 3) 6 = Problem continues on next page. July 2013 Page 14 of 33

15 Problem continued from previous page. 5 0 = = = = = = = = 55 2 = 5 3 = = 23 5 = 2 2 = = 1 4 July 2013 Page 15 of 33

16 SE TASK: Nesting Dolls The first Russian nesting dolls were made over a hundred years ago. They were brightly painted in classic Russian designs and made so that the finished doll had many similar dolls that fit one inside the other. This type of doll is called a Matroyshka which is derived from the Latin word for mother and is a common name for women in Russian villages. Suppose there are a total of eight dolls. The finished doll is n centimeters tall with seven smaller dolls that fit inside one another. Each doll is 3 2 the height of the next larger doll. a. What is the relationship of the heights of the third largest doll and the largest doll? b. What is the relationship of the smallest doll and the largest doll? As Russia has dropped the Iron Curtain and opened doors, many of their production facilities have more technology. Instead of hand carving the Matroyshka, factories use computerized robots to make them. You have been asked to develop the general statement to generalize the height of each doll regardless of the number of dolls in the finished product. July 2013 Page 16 of 33

17 SE Task: Exponential Exponents Write your numeric expression in the box in the center of the page. With your group, use the properties of integer exponents to generate expressions equivalent to your numeric expression. Each property of exponents should be represented. Write an explanation of the strategies that you used to generate the equivalent expressions. July 2013 Page 17 of 33

18 SE TASK: Exploring Powers of 10 This activity will help you to learn how to represent very large and very small numbers. Part 1: Very Small Numbers (negative exponents) Complete the following table. What patterns do you see? 10 6 = 10 5 = 10 4 = 10 3 = 10 2 = 10 1 = 10 0 = 10-1 = 10-2 = 10-3 = July 2013 Page 18 of 33

19 DIRECTIONS FOR USING THE TI-83 or TI-84: Exploring Powers of 10 It may be appropriate to use a calculator or computer when data may contain numbers that are difficult to work with using paper and pencil. Exploring very large and very small numbers can be done on the TI-83 or TI-84 graphing calculator with the following instructions. 1. Turn on the calculator ON. ^ 2. Enter the number 10, press the key, enter the number 6, and press ENTER. 3. Continue step 2 for the positive exponents and zero. 4. Enter the number 10, press the key, press the negation ( ) key, enter the number 1, and press ENTER. ^ 5. Continue step 4 for the negative exponents. July 2013 Page 19 of 33

20 Part 2: Very Large Numbers (positive exponents) Sometimes it can be cumbersome to say and write numbers in their common form (for example trying to display numbers on a calculator with limited screen space). Another option is to use exponential notation and the base-ten place-value system. Using a calculator (with exponential capabilities), complete each of the following: 1. Explore 10 N for various values of N. What patterns do you notice? 2. Enter 45 followed by a string of zeros. How many will your calculator permit? What happens when you press enter? What does 4.5E10 mean? What about 2.3E4? Can you enter this another way? 3. Try sums like ( N ) + (2 10 K ) for different values of N and K. Describe any patterns that you may notice. 4. Try products like ( N ) (2 10 K ). Describe any patterns that you may notice. 5. Try quotients like ( N ) (2 10). Describe any patterns that you may notice. July 2013 Page 20 of 33

21 Short Cycle Task: Ponzi Pyramid Scheme Source: Balanced Assessment Materials from Mathematics Assessment Project letter. In this task, students will experience a real-world application of exponents using a chain STANDARDS ADDRESSED IN THIS TASK: Work with radicals and integer exponents. MCC8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, ( 5) = 3 ( 3) 1 = = MCC8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 =p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. MCC8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as and the population of the world as , and determine that the world population is more than 20 times larger. MCC8.EE.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. STANDARDS FOR MATHEMATICAL PRACTICE: This task uses all of the practices with emphasis on: 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 6. Attend to precision. 7. Look for and make use of structure. July 2013 Page 21 of 33

22 TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION: Tasks and lessons from the Mathematics Assessment Project are specifically designed to help teachers effectively formatively assess their students. The way the tasks and lessons are designed gives the teacher a clear understanding of what the students are able to do and not do. Within the lesson, teachers will find suggestions and question prompts that will help guide students towards understanding. For more information access the MAP website: The task, Ponzi Pyramid Schemes, is a Mathematics Assessment Project Assessment Task that can be found at the website: The PDF version of the task can be found at the link below: The scoring rubric can be found at the following link: July 2013 Page 22 of 33

23 Formative Assessment Lesson: Estimating Length Using Scientific Notation Source: Formative Assessment Lesson Materials from Mathematics Assessment Project Estimating lengths of everyday objects, converting between decimal and scientific notation, and making comparison of the size of numbers expressed in both decimal and scientific notation. STANDARDS ADDRESSED IN THIS TASK: MCC8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as and the population of the world as , and determine that the world population is more than 20 times larger. MCC8.EE.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. STANDARDS FOR MATHEMATICAL PRACTICE: This task uses all of the practices with emphasis on: 2. Reason abstractly and quantitatively. 7. Look for and make use of structure. TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION Tasks and lessons from the Mathematics Assessment Project are specifically designed to help teachers effectively formatively assess their students. The way the tasks and lessons are designed gives the teacher a clear understanding of what the students are able to do and not do. Within the lesson, teachers will find suggestions and question prompts that will help guide students towards understanding. For more information access the MAP website: The task, Estimating Length Using Scientific Notation, is a Formative Assessment Lesson (FAL) that can be found at the website: The FAL document provides a clear lesson design, from the opening of the lesson to the closing of the lesson. July 2013 Page 23 of 33

24 The PDF version of the task can be found at the link below: July 2013 Page 24 of 33

25 SE TASK: E. coli The Center for Disease Control (CDC) monitors public facilities to assure that there are no health risks to the people that visit these facilities. Occasionally, Escherichia coli will be discovered, which is also known as E. coli. This is a type of bacterium that can make people very sick. The mass of each E. coli bacterium is gram. The last time that E. coli was discovered in Georgia, the number of bacteria had been increasing for 24 hours! This caused each bacterium to be replaced by a population of bacteria. a. What was the mass of the population of E. coli when it was discovered? Justify your answer. b. If the mass of a small paper clip is about 1 gram, how many times more is its mass than that of the discovered E. coli bacteria? Show how you know. Visit for more information on E. coli. July 2013 Page 25 of 33

26 Short Cycle Task: Giantburgers Source: Mathematics Assessment Project Shell Center/MARS, University of Nottingham & UC Berkeley In this task, the student will use properties of exponents and scientific notation. STANDARDS ADDRESSED IN THIS TASK: Work with radicals and integer exponents. MCC8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, ( 5) = 3 ( 3) 1 = = MCC8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as and the population of the world as , and determine that the world population is more than 20 times larger. MCC8.EE.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. STANDARDS FOR MATHEMATICAL PRACTICE: This task uses all of the practices with emphasis on: 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 6. Attend to precision. 7. Look for and make use of structure. July 2013 Page 26 of 33

27 SE TASK: Giantburgers This headline appeared in a newspaper. Decide whether this headline is true using the following information. There are about Giantburger restaurants in America. Each restaurant serves about people every day. There are about Americans. Explain your reasons and show clearly how you figured it out. July 2013 Page 27 of 33

28 Short Cycle: 100 People Source: Balanced Assessment Materials from Mathematics Assessment Project In this task, students will use scientific notation in a real-world application of using population. STANDARDS ADDRESSED IN THIS TASK: Work with radicals and integer exponents. MCC8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, ( 5) = 3 ( 3) 1 = = MCC8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 =p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. MCC8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as and the population of the world as , and determine that the world population is more than 20 times larger. MCC8.EE.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. STANDARDS FOR MATHEMATICAL PRACTICE: This task uses all of the practices with emphasis on: 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 6. Attend to precision. 7. Look for and make use of structure. July 2013 Page 28 of 33

29 TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION: Tasks and lessons from the Mathematics Assessment Project are specifically designed to help teachers effectively formatively assess their students. The way the tasks and lessons are designed gives the teacher a clear understanding of what the students are able to do and not do. Within the lesson, teachers will find suggestions and question prompts that will help guide students towards understanding. For more information access the MAP website: The task, 100 People, is a Mathematics Assessment Project Assessment Task that can be found at the website: The PDF version of the task can be found at the link below: The scoring rubric can be found at the following link: July 2013 Page 29 of 33

30 Formative Assessment Lesson: Solving Linear Equations in One Variable Source: Formative Assessment Lesson Materials from Mathematics Assessment Project This lesson unit is intended to help you assess how well students are able to: Solve linear equations in one variable with rational number coefficients. Collect like terms. Expand expressions using the distributive property. Categorize linear equations in one variable as having one, none, or infinitely many solutions. STANDARDS ADDRESSED IN THIS TASK: Understand the connections between proportional relationships, lines, and linear equations. MCC8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. MCC8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Define, evaluate, and compare functions. MCC8.F.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. STANDARDS FOR MATHEMATICAL PRACTICE: This task uses all of the practices with emphasis on: 1. Make sense of problems and persevere in solving them 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure. July 2013 Page 30 of 33

31 TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION: Tasks and lessons from the Mathematics Assessment Project are specifically designed to help teachers effectively formatively assess their students. The way the tasks and lessons are designed gives the teacher a clear understanding of what the students are able to do and not do. Within the lesson, teachers will find suggestions and question prompts that will help guide students towards understanding. For more information access the MAP website: The task, Solving Linear Equations in One Variable, is a Formative Assessment Lesson (FAL) that can be found at the website: The FAL document provides a clear lesson design, from the opening of the lesson to the closing of the lesson. The PDF version of the task can be found at the link below: July 2013 Page 31 of 33

32 SE TASK: Writing for a Math Website As a community service project your school has partnered with a national web based company to help develop a website that will provide math support to 8 th graders with the Common Core Georgia Performance Standards. Your middle school has been assigned the standards below. MCC8.EE.7 Solve linear equations in one variable. a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = b, or a = a results (where a and b are different numbers). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Your help has been requested writing the problems and providing the solutions. The new program will be interactive. It must tell the students if the answer is correct or incorrect and show the problems completely worked out along with justification for each step. 1. Create 6 equations that can be sorted into the categories in the table below (2 per category) and provide solutions to the problem. The solutions must show all steps along with written justifications for each step. Each problem created must meet the specs of the standard as listed below: linear equation in one variable include rational number coefficients require use of the distributive property variables on both sides of the equation require collecting like terms July 2013 Page 32 of 33

33 One Solution No Solution Infinitely Many Solutions 2. Provide a written explanation to help other 8 th graders understand the meaning of having one solution, no solution, or infinitely many solutions. July 2013 Page 33 of 33