ࡱ> x_ Mbjbj pbbc< ff8LNT&&&!!!!!!$E)&^&))Eff 3.3.3.)f83.)3.3.FaPS@/7*T $0Td<+r<aa<&hT'J3.'<'?&&&EE,&&&T))))<&&&&&&&&& ,: Rigorous Curriculum Design Unit Planning Organizer Subject(s)MathGrade/Course8thUnit of StudyUnit 1: The Number System & ExponentsUnit Type(s)Q'Topical X Skills-based Q' Thematic Pacing20 days Unit Abstract  In this unit of study, students will apply the properties of exponents. They will represent very small or very large numbers in scientific notation, perform operations and learn how to interpret when E appears on the calculator. Students will provide examples of linear equations with one, infinitely many, or no solution. Students will understand that real numbers are rational or irrational; will place them on the number line and compare them.  Common Core Essential State Standards  Domains: Expressions and Equations (8.EE), Number System (8.NS) Clusters: Work with radicals and integer exponents. Analyze and solve linear equations. Know that there are numbers that are not rational, and approximate them by rational numbers. Standards: 8.EE.1 KNOW and APPLY the properties of integer exponents to GENERATE equivalent numerical expressions. For example: 32 35 = 33 = 1/33 = 1/27. 8.EE.2 USE square root and cube root symbols to REPRESENT solutions to equations of the form x = p and x = 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. 8.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 3 108 and the population of the world as 7 109, and determine that the world population is more than 20 times larger. 8.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. 8.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 = a, or a = b results (where a and b are different numbers). b. Solve linear equations with rational number coefficients, including equation whose solutions require expanding expressions using the distributive property and collecting like terms. 8.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. 8.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., 2). For example, by truncating the decimal expansion 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  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.  Unpacked Standards 8.EE.1 In 6th grade, students wrote and evaluated simple numerical expressions with whole number exponents (i.e. 53 = 5 5 5 = 125). Integer (positive and negative) exponents are further developed to generate equivalent numerical expressions when multiplying, dividing or raising a power to a power. Using numerical bases and the laws of exponents, students generate equivalent expressions. Students understand: Bases must be the same before exponents can be added, subtracted or multiplied. (Example 1) Exponents are subtracted when like bases are being divided (Example 2) A number raised to the zero (0) power is equal to one. (Example 3) Negative exponents occur when there are more factors in the denominator. These exponents can be expressed as a positive if left in the denominator. (Example 4) Exponents are added when like bases are being multiplied (Example 5) Exponents are multiplied when an exponents is raised to an exponent (Example 6) Several properties may be used to simplify an expression (Example 7) Example 1:  =  Example 2:  =  =  =  =  EMBED Equation.3  Example 3: 60 = 1 Students understand this relationship from examples such as . This expression could be simplified as  EMBED Equation.3  = 1. Using the laws of exponents this expression could also be written as 62-2 = 60. Combining these gives 60 = 1. Example 4: =  x  =  x  =  x  =  Example 5: (32) (34) = (32+4) = 36 = 729 Example 6: (43)2 = 43x2 = 46 = 4,096 Example 7:      8.EE.2 Students recognize perfect squares and cubes, understanding that non-perfect squares and non-perfect cubes are irrational. Students 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. Example 1: 4 = 16 and "16 = 4 NOTE: (-4)= 16 while -4 = -16 since the negative is not being squared. This difference is often problematic for students, especially with calculator use. Example 2:  Note: there is no negative cube root since multiplying 3 negatives would give negative. This understanding is used to solve equations containing square or cube numbers. Rational numbers would have perfect squares or perfect cubes for the numerator and denominator. In the standard, the value of p for square root and cube root equations must be positive. Example 3: Solve: x2 = 25 Solution:  =  x = 5 NOTE: There are two solutions because 5 5 and -5 -5 will both equal 25. Example 4: Solve: x2 =  Solution:  =  x =  Example 5: Solve: x3 = 27 Solution:  EMBED Equation.3  =  x = 3 Example 6: Solve: x3 =  Solution:  EMBED Equation.3  =  x =  Students understand that in geometry the square root of the area is the length of the side of a square and a cube root of the volume is the length of the side of a cube. Students use this information to solve problems, such as finding the perimeter. Example 7: What is the side length of a square with an area of 49 ft2? Solution:  = 7 ft. The length of one side is 7 ft. 8.EE.3 Students use scientific notation to express very large or very small numbers. Students compare and interpret scientific notation quantities in the context of the situation, recognizing that if the exponent increases by one, the value increases 10 times. Likewise, if the exponent decreases by one, the value decreases 10 times. Students solve problems using addition, subtraction or multiplication, expressing the answer in scientific notation. Example 1: Write 75,000,000,000 in scientific notation. Solution: 7.5 x 1010 Example 2: Write 0.0000429 in scientific notation. Solution: 4.29 x 10-5 Example 3: Express 2.45 x 105 in standard form. Solution: 245,000 Example 4: How much larger is 6 x 105 compared to 2 x 103 Solution: 300 times larger since 6 is 3 times larger than 2 and 105 is 100 times larger than 103. Example 5: Which is the larger value: 2 x 106 or 9 x 105? Solution: 2 x 106 because the exponent is larger 8.EE.4 Students understand scientific notation as generated on various calculators or other technology. Students enter scientific notation using E or EE (scientific notation), * (multiplication), and ^ (exponent) symbols. Example 1: 2.45E+23 is 2.45 x 1023 and 3.5E-4 is 3.5 x 10-4 NOTE: There are other notations for scientific notation depending on the calculator being used. Students add and subtract with scientific notation. Example 2: In July 2010 there were approximately 500 million facebook users. In July 2011 there were approximately 750 million facebook users. How many more users were there in 2011? Write your answer in scientific notation. Solution: Subtract the two numbers: 750,000,000 - 500,000,000 = 250,000,000 ! 2.5 x 108 Students use laws of exponents to multiply or divide numbers written in scientific notation, writing the product or quotient in proper scientific notation. Example 3: (6.45 x 1011)(3.2 x 104) = (6.45 x 3.2)(1011 x 104) Rearrange factors = 20.64 x 1015 Add exponents, multiplying powers of 10 = 2.064 x 1016 Write in scientific notation Example 4: 3.45 x 105 = 3.45 x 105 (-2) Subtract exponents when dividing powers of 6.7 x 10-2 6.7 = 0.515 x 107 Write in scientific notation = 5.15 x 106 Example 5: (0.0025)(5.2 x 104) = (2.5 x 10-3)(5.2 x 105) Write factors in scientific notation = (2.5 x 5.2)(10-3 x 105) Rearrange factors = 13 x 10 2 Add exponents when multiplying powers of 10 = 1.3 x 103 Write in scientific notation Example 6: The speed of light is 3 x 10 8 meters/second. If the sun is 1.5x 1011 meters from earth, how many seconds does it take light to reach the earth? Express your answer in scientific notation. Solution: 5 x 102 (light)(x) = sun, where x is the time in seconds (3 x 10 8 )x = 1.5 x 1011 1.5 x 1011 3 x 10 8 Students understand the magnitude of the number being expressed in scientific notation and choose an appropriate corresponding unit. Example 7: 3 x 108 is equivalent to 300 million, which represents a large quantity. Therefore, this value will affect the unit 8.EE.7 Students solve one-variable equations including those with the variables being on both sides of the equals sign. Students recognize that the solution to the equation is the value(s) of the variable, which make a true equality when substituted back into the equation. Equations shall include rational numbers, distributive property and combining like terms. Example 1: Equations have one solution when the variables do not cancel out. For example, 10x 23 = 29 3x can be solved to x = 4. This means that when the value of x is 4, both sides will be equal. If each side of the equation were treated as a linear equation and graphed, the solution of the equation represents the coordinates of the point where the two lines would intersect. In this example, the ordered pair would be (4, 17). 10 4 23 = 29 3 4 40 23 = 29 12 17 = 17 Example 2: Equations having no solution have variables that will cancel out and constants that are not equal. This means that there is not a value that can be substituted for x that will make the sides equal. -x + 7 6x = 19 7x Combine like terms -7x + 7 = 19 7x Add 7x to each side 7 `" 19 This solution means that no matter what value is substituted for x the final result will never be equal to each other. If each side of the equation were treated as a linear equation and graphed, the lines would be parallel. Example 3: An equation with infinitely many solutions occurs when both sides of the equation are the same. Any value of x will produce a valid equation. For example the following equation, when simplified will give the same values on both sides. (36a 6) = (4 24a) 18a + 3 = 3 18a If each side of the equation were treated as a linear equation and graphed, the graph would be the same line. Students write equations from verbal descriptions and solve. Example 4: Two more than a certain number is 15 less than twice the number. Find the number. Solution: n + 2 = 2n 15 17 = n 8.NS.1 Students understand that Real numbers are either rational or irrational. They distinguish between rational and irrational numbers, recognizing that any number that can be expressed as a fraction is a rational number. The diagram below illustrates the relationship between the subgroups of the real number system.  Students recognize that the decimal equivalent of a fraction will either terminate or repeat. Fractions that terminate will have denominators containing only prime factors of 2 and/or 5. This understanding builds on work in 7th grade when students used long division to distinguish between repeating and terminating decimals. Students convert repeating decimals into their fraction equivalent using patterns or algebraic reasoning. One method to find the fraction equivalent to a repeating decimal is shown below. Example 1: Change 0. to a fraction. Let x = 0.444444.. Multiply both sides so that the repeating digits will be in front of the decimal. In this example, one digit repeats so both sides are multiplied by 10, giving 10x = 4.4444444. Subtract the original equation from the new equation. 10x = 4.4444444. x = 0.444444. 9x = 4 Solve the equation to determine the equivalent fraction. 9x = 4 9 9 x = 4 9 Additionally, students can investigate repeating patterns that occur when fractions have denominators of 9, 99, or 11. Example 2:  EMBED Equation.3  is equivalent to 0. EMBED Equation.3 ,  EMBED Equation.3  is equivalent to 0. EMBED Equation.3 , etc. 8.NS.2 Students locate rational and irrational numbers on the number line. Students compare and order rational and irrational numbers. Students also recognize that square roots may be negative and written as - "28. Example 1: Compare  EMBED Equation.3  and  EMBED Equation.3  .  Solution: Statements for the comparison could include:  EMBED Equation.3  and  EMBED Equation.3  are between the whole numbers 1 and 2  EMBED Equation.3  is between 1.7 and 1.8  EMBED Equation.3  is less than  EMBED Equation.3  Additionally, students understand that the value of a square root can be approximated between integers and that nonperfect square roots are irrational. Example 2: Find an approximation of  EMBED Equation.3  Determine the perfect squares  EMBED Equation.3 is between, which would be 25 and 36. The square roots of 25 and 36 are 5 and 6 respectively, so we know that  EMBED Equation.3  is between 5 and 6. Since 28 is closer to 25, an estimate of the square root would be closer to 5. One method to get an estimate is to divide 3 (the distance between 25 and 28) by 11 (the distance between the perfect squares of 25 and 36) to get 0.27. The estimate of  EMBED Equation.3  would be 5.27 (the actual is 5.29)  Unpacked Concepts (students need to know) Unwrapped Skills (students need to be able to do) Cognition (DOK) 8.EE.1 Properties of exponents  I can use properties of exponents to simplify expressions 2  8EE.2 Perfect squares & cubes I can solve and explain equations in the form of x2 = p and x3 = p 28.EE.3 Scientific notation  I can represent very small or very large numbers in scientific notation. I can compare quantities written in scientific notation. 2 38.EE.4 Computation in scientific notation and decimal form Scientific notation on calculator  I can compare and compute numbers in scientific notation and decimal form. I can interpret how to read answer when the E appears con the calculator. 2 28.EE.7 Linear equations with one, infinitely many, or no solution  I can provide examples of linear equations in one variable with one, infinitely many or no solutions. I can solve equations that include rational number coefficients, expanding expressions, and combining like terms. 3 2 8.NS.1 Rational or irrational numbers I can classify numbers as rational or irrational and explain why.  2  8.NS.2 Comparison of irrational numbers  I can use the knowledge of square roots of perfect squares to estimate value of other square roots. I can order rational and irrational numbers on number line. I can find approximate location of irrational numbers on a number line. 2 2 2 Essential Questions Corresponding Big Ideas8.EE.1 How can I apply properties of exponents to simplify expressions?  Students will apply laws of exponents to write equivalent expressions for a given expression. 8.EE.2 How can I solve equations of the form x2 = p? How can I solve equations of the form x3 = p?  Students will solve x2 = p, where p is a positive rational number represent the solution using the square root symbol, and can explain when and why the solution must include the (plus or minus) symbol. Students will solve x3 = p, where p is a positive rational number and represent the solution using the cube root symbol. The student knows that the cube root of a positive number is positive, and the cube root of a negative number is negative.8EE.3 How can I represent very large or very small numbers in scientific notation? How can I compare two quantities written in scientific notation?  Students will represent very large and very small quantities/measurements in scientific notation. Students will compare two quantities written in scientific notation and can reason how many times bigger/smaller one is than the other without having to convert each number back into decimal form.8.EE.4 How can I solve problems where both decimal and scientific notation are used? How can I interpret the answer when E appears on the calculator?  Students will solve problems that involve very large or very small quantities where both decimal and scientific notation are used. Students will interpret how to read the answer when the E appears when using a calculator to perform a calculation in which the answer will be a very large or very small number. 8.EE.7 How can I provide examples of linear equations with one, infinitely many or no solution? How can I solve equations that include rational number coefficients, expanding expressions, and combining like terms? Students will provide an example of a linear equation in one variable that has exactly one solution, infinitely many solutions, or no solution. Students will solve linear equations that include rational number coefficients, expanding expressions and combining like terms. 8.NS.1 How can I classify numbers as rational or irrational and explain? Students will classify numbers as rational or irrational and explain why. 8.NS.2 How can I use knowledge of square roots of perfect squares to estimate the value of other squares? How can I order and compare rational or irrational from least to greatest? How can I approximate the location of irrational numbers on the number line?  Students will estimate value of square root of non perfect squares. Students will order rational and irrational numbers. Students will approximate location of irrational numbers on the number line. Vocabulary  laws of exponents, power, perfect squares, perfect cubes, root, square root, cube root, scientific notation, standard form of a number, intersecting, parallel lines, coefficient, distributive property, like terms, real numbers, irrational numbers, rational numbers, integers, whole numbers, natural numbers, radical, radicand, terminating decimals, repeating decimals, truncate Language ObjectivesKey Vocabulary 8.EE.1 - 4 8.EE.7 Define and give examples of vocabulary and expressions specific to this standard (properties, integer exponents, positive, negative, equivalent numerical expressions, raising to a power, square root, cube root, squaring, cubing, rational, irrational, inverse operations, scientific notation, distributive property, rational numbers, variables, equality, equation, solution, like terms, constant, value, linear equation, expanding expressions, coefficients, etc.) Language Function 8.EE.7 SWBAT use sentence frames to explain why some equations have no solution. 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