Algebra I - Unpacked Contents

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p. 1

the real number system common core cluster extend the properties of exponents to rational exponents common core standard n-rn.1 explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values allowing for a notation for radicals in terms of rational exponents for example we define to be the cube root of 5 because we want to hold so n-rn unpacking what does this standard mean that a student will know and be able to do n-rn.1 in order to understand the meaning of rational exponents students can initially investigate them by considering a pattern such as what is the pattern for the exponents they are reduced by a factor of each time what is the pattern of the simplified values each successive value is the square root of the previous value if we continue this pattern then once the meaning of a rational exponent with a numerator of 1 is established students can verify that the properties of integer exponents hold for rational exponents as well for example must equal 5 since since ex use an example to show why holds true for expressions involving rational exponents like or !

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p. 2

n-rn.2 rewrite expressions involving radicals and rational exponents using the properties of exponents n-rn.2 students should be able to use the properties of exponents to rewrite expressions involving radicals as expressions using rational exponents at this level focus on fractional exponents with a numerator of 1 ex simplify the following a b n-rn.2 students should be able to use the properties of exponents to rewrite expressions involving rational exponents as expressions using radicals at this level focus on fractional exponents with a numerator of 1 ex simplify the following a b !

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p. 3

quantities common core cluster reason quantitatively and use units to solve problems common core standard n-q.1 use units as a way to understand problems and to guide the solution of multi-step problems choose and interpret units consistently in formulas choose and interpret the scale and the origin in graphs and data displays n-q unpacking what does this standard mean that a student will know and be able to do n-q.1 use units as a tool to help solve multi-step problems for example students should use the units assigned to quantities in a problem to help identify which variable they correspond to in a formula students should also analyze units to determine which operations to use when solving a problem given the speed in mph and time traveled in hours what is the distance traveled from looking at the units we can determine that we must multiply mph times hours to get an answer expressed in miles note that knowledge of the distance formula is not required to determine the need to multiply in this case n-q.1 based on the type of quantities represented by variables in a formula choose the appropriate units to express the variables and interpret the meaning of the units in the context of the relationships that the formula describes ex when finding the area of a circle using the formula which unit of measure would be appropriate for the radius a square feet b inches c cubic yards d pounds ex based on your answer to the previous question what units would the area be measured in n-q.1 when given a graph or data display read and interpret the scale and origin when creating a graph or data display choose a scale that is appropriate for viewing the features of a graph or data display understand that using larger values for the tick marks on the scale effectively zooms out from the graph and choosing smaller values zooms in understand that the viewing window does not necessarily show the x or y-axis but the apparent axes are parallel to the x and y-axes hence the intersection of the apparent axes in the viewing window may not be the origin also be aware that apparent intercepts may not correspond to the actual x or y-intercepts of the graph of a function

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p. 4

ex what scale would be appropriate for making a histogram of the following data that describes the level of lead in the blood of children in micrograms per deciliter who were exposed to lead from their parents workplace 10 13 14 15 16 17 18 20 21 22 23 23 24 25 27 31 34 34 35 35 36 37 38 39 39 41 43 44 45 48 49 62 73 n-q.2 define appropriate quantities for the purpose of descriptive modeling n-q.2 define the appropriate quantities to describe the characteristics of interest for a population for example if you want to describe how dangerous the roads are you may choose to report the number of accidents per year on a particular stretch of interstate generally speaking it would not be appropriate to report the number of exits on that stretch of interstate to describe the level of danger ex what quantities could you use to describe the best city in north carolina ex what quantities could you use to describe how good a basketball player is n-q.3 choose a level of accuracy appropriate to limitations on measurement when reporting quantities n-q.3 understand that the tool used determines the level of accuracy that can be reported for a measurement for example when using a ruler you can only legitimately report accuracy to the nearest division if i use a ruler that has centimeter divisions to measure the length of my pencil i can only report its length to the nearest centimeter ex what is the accuracy of a ruler with 16 divisions per inch?

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p. 5

seeing structure in expressions common core cluster interpret the structure of expressions common core standard a-sse.1 interpret expressions that represent a quantity in terms of its context a interpret parts of an expression such as terms factors and coefficients b interpret complicated expressions by viewing one or more of their parts as a single entity for example interpret as the product of p and a factor not depending on p a-sse unpacking what does this standard mean that a student will know and be able to do a-sse.1a students manipulate the terms factors and coefficients in difficult expressions to explain the meaning of the individual parts of the expression use them to make sense of the multiple factors and terms of the expression for example consider the expression 10,0001.0555 this expression can be viewed as the product of 10,000 and 1.055 raised to the 5th power 10,000 could represent the initial amount of money i have invested in an account the exponent tells me that i have invested this amount of money for 5 years the base of 1.055 can be rewritten as 1 0.055 revealing the growth rate of 5.5 per year at this level limit to linear expressions exponential expressions with integer exponents and quadratic expressions ex the expression 204x 500 represents the cost in dollars of the materials and labor needed to build a square fence with side length x feet around a playground interpret the constants and coefficients of the expression in context a-sse.1b students group together parts of an expression to reveal underlying structure for example consider the expression that represents income from a concert where p is the price per ticket the equivalent factored form shows that the income can be interpreted as the price times the number of people in attendance based on the price charged at this level limit to linear expressions exponential expressions with integer exponents and quadratic expressions ex without expanding explain how the expression can be viewed as having the structure of a quadratic expression a.sse.2 students rewrite algebraic expressions by combining like terms or factoring to reveal equivalent forms of the same expression ex expand the expression to show that it is a quadratic expression of the form a-sse.2 use the structure of an expression to identify ways to rewrite it for example see x4 ­ y4 as x22 ­ y22 thus recognizing it as a difference of squares that can be factored as x2 ­ y2 x2 y2

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p. 6

seeing structure in expressions common core cluster write expressions in equivalent forms to solve problems common core standard a-sse.3 choose and produce an a-sse unpacking what does this standard mean that a student will know and be able to do a-sse.3a students factor quadratic expressions and find the zeros of the quadratic function they represent zeroes are the x-values that yield a y-value of 0 students should also explain the meaning of the zeros as they relate to the problem for example if the expression x2 ­ 4x 3 represents the path of a ball that is thrown from one person to another then the expression x ­ 1 x ­ 3 represents its equivalent factored form the zeros of the function x ­ 1 x ­ 3 y would be x 1 and x 3 because an x-value of 1 or 3 would cause the value of the function to equal 0 this also indicates the ball was thrown after 1 second of holding the ball and caught by the other person 2 2 seconds later at this level limit to quadratic expressions of the form ax bx c ex the expression is the income gathered by promoters of a rock concert based on the ticket price m for what values of m would the promoters break even equivalent form of an expression to reveal and explain properties of the quantity represented by the expression a factor a quadratic expression to reveal the zeros of the function it defines !

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p. 7

arithmetic with polynomials and rational expressions common core cluster perform arithmetic operations on polynomials common core standard a-apr.1 understand that a-apr unpacking what does this standard mean that a student will know and be able to do a-apr.1 the closure property means that when adding subtracting or multiplying polynomials the sum difference or product is also a polynomial polynomials are not closed under division because in some cases the result is a rational expression rather than a polynomial at this level limit to addition and subtraction of polynomials form a system analogous to the integers namely they are closed under the operations of addition subtraction and multiplication add subtract and multiply polynomials quadratics and multiplication of linear expressions a-apr.1 add subtract and multiply polynomials at this level limit to addition and subtraction of quadratics and multiplication of linear expressions ex if the radius of a circle is kilometers what would the area of the circle be ex explain why !does not equal creating equations common core cluster create equations that describe numbers or relationships common core standard unpacking what does this standard mean that a student will know and be able to do a-ced

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p. 8

a-ced.1 create equations and inequalities in one variable and use them to solve problems include equations arising from linear and quadratic functions and simple rational and exponential functions a-ced.1 from contextual situations write equations and inequalities in one variable and use them to solve problems include linear and exponential functions at this level focus on linear and exponential functions ex the tindell household contains three people of different generations the total of the ages of the three family members is 85 a find reasonable ages for the three tindells b find another reasonable set of ages for them c one student in solving this problem wrote c c+20 c+56 85 1 what does c represent in this equation 2 what do you think the student had in mind when using the numbers 20 and 56 3 what set of ages do you think the student came up with ex a salesperson earns $700 per month plus 20 of sales write an equation to find the minimum amount of sales needed to receive a salary of at least $2500 per month ex a scientist has 100 grams of a radioactive substance half of it decays every hour write an equation to find how long it takes until 25 grams are left a-ced.2 create equations in two or more variables to represent relationships between quantities graph equations on coordinate axes with labels and scales a-ced.2 given a contextual situation write equations in two variables that represent the relationship that exists between the quantities also graph the equation with appropriate labels and scales make sure students are exposed to a variety of equations arising from the functions they have studied at this level focus on linear exponential and quadratic equations limit to situations that involve evaluating exponential functions for integer inputs ex in a woman s professional tennis tournament the money a player wins depends on her finishing place in the standings the first-place finisher wins half of $1,500,000 in total prize money the second-place finisher wins half of what is left then the third-place finisher wins half of that and so on a write a rule to calculate the actual prize money in dollars won by the player finishing in nth place for any positive integer n b graph the relationship that exists between the first 10 finishers and the prize money in dollars c what pattern do you notice in the graph what type of relationship exists between the two variables?

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p. 9

a-ced.3 represent constraints by equations or inequalities and by systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context for example represent inequalities describing nutritional and cost constraints on combinations of different foods a-ced.3 use constraints which are situations that are restricted to develop equations and inequalities and systems of equations or inequalities describe the solutions in context and explain why any particular one would be the optimal solution limit to linear equations and inequalities ex the elite dance studio budgets a maximum of $100 per month for newspaper and yellow pages advertising the news paper charges $50 per ad and requires at least four ads per month the phone company charges $100 dollars for half a page and requires a minimum of two advertisements per month it is estimated that each newspaper ad reaches 8000 people and that each half page of yellow page advertisement reaches 15,000 people what combination of newspaper and yellow page advertising should the elite dance studio use in order to reach the maximum number of people what assumptions did you make in solving this problem how realistic do you think they are a-ced.4 solve multi-variable formulas or literal equations for a specific variable explicitly connect this to the process of solving equations using inverse operations limit to formulas which are linear in the variable of a-ced.4 rearrange formulas to highlight a quantity of interest using the same reasoning as in solving equations for example rearrange ohm s law v ir to highlight resistance r interest or to formulas involving squared or cubed variables ex if h ka t1 t2 solve for t2 l reasoning with equations and inequalities common core cluster understanding solving equations as a process of reasoning and explain the reasoning common core standard unpacking what does this standard mean that a student will know and be able to do a-rei

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p. 10

a-rei.1 explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step starting from the assumption that the original equation has a solution construct a viable argument to justify a solution method a-rei.1 relate the concept of equality to the concrete representation of the balance of two equal quantities properties of equality are ways of transforming equations while still maintaining equality/balance assuming an equation has a solution construct a convincing argument that justifies each step in the solution process with mathematical properties ex solve 5x+3 3x=55 for x use mathematical properties to justify each step in the process common core cluster solve equations and equalities in one variable common core standard a-rei.3 solve linear equations and unpacking what does this standard mean that a student will know and be able to do a-rei.3 solve linear equations in one variable including coefficients represented by letters ex solve ax +b =c for x what are the specific restrictions on a ex what is the difference between solving an equation and simplifying an expression ex grandma s house is 20 miles away and johnny wants to know how long it will take to get there using various modes of transportation a model this situation with an equation where time is a function of rate in miles per hour b for each mode of transportation listed below determine the time it would take to get to grandma s mode of transportation rate of travel in mph time of travel hrs bike car walking 12mph 55mph 4mph inequalities in one variable including equations with coefficients represented by letters a-rei.3 solve linear inequalities in one variable including coefficients represented by letters ex a parking garage charges $1 for the first half-hour and $0.60 for each additional half-hour or portion thereof if you have only $6.00 in cash write an inequality and solve it to find how long you can park ex compare solving an inequality in one variable to solving an equation in one variable also compare solving a linear inequality to solving a linear equation.

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p. 11

common core cluster solve systems of equations common core standard a-rei.6 solve systems of linear equations exactly and approximately e.g with graphs focusing on pairs of linear equations in two variables unpacking what does this standard mean that a student will know and be able to do a.rei.6 solve systems of equations exactly by using the substitution method and solve systems of equations by using the elimination method sometimes called linear combinations ex solve the system by elimination checking your solution by graphing using technology 3x 2y 6 x 4y 2 ex solve the system by substitution checking your solution by graphing using technology -3x 5y 6 2x y 6 a.rei.6 solve systems of equations approximately by using graphs graph the system of linear functions on the same coordinate plane and find the point of intersection this point is the solution to the system because it is the one point that makes all equations in the system true equations may be in standard or slope-intercept form ex the equations y 18 .4m and y 11.2 .54m give the lengths of two different springs in centimeters as mass is added in grams m to each separately a graph each equation on the same set of axes b what mass makes the springs the same length c what is the length at that mass d write a sentence comparing the two springs common core cluster represent and solve equations and inequalities graphically common core standard unpacking what does this standard mean that a student will know and be able to do?

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p. 12

a-rei.10 understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane often forming a curve which could be a line a-rei.10 understand that all points on the graph of a two-variable equation are solutions because when substituted into the equation they make the equation true at this level focus on linear and exponential equations ex which of the following points are on the graph of the equation how many points are on this graph explain a 4 0 b 0 10 c 1 7.5 d 2.3 5 a-rei.11 understand that solving a one-variable equation of the form fx gx is the same as solving the twovariable system y fx and y gx when solving by graphing the x-values of the intersection points of y fx and y gx is the solution of fx gx for any combination of linear and exponential functions use technology entering fx in y1 and gx in y2 graphing the equations to find their point of equality at this level a-rei.11 explain why the xcoordinates of the points where the graphs of the equations y fx and y gx intersect are the solutions of the equation fx gx find the solutions approximately e.g using technology to graph the functions make tables of values or find successive approximations include cases where fx and/or gx are linear polynomial rational absolute value exponential and logarithmic functions focus on linear and exponential functions ex how do you find the solution to an equation graphically a-rei.11 solve graphically finding approximate solutions using technology at this level focus on linear and exponential functions ex solve the following equations by graphing give your answer to the nearest tenth 10x +5 -x +8 a-rei.11 solve by making tables for each side of the equation use the results from substituting previous values of x to decide whether to try a larger or smaller value of x to find where the two sides are equal the x-value that makes the two sides equal is the solution to the equation at this level focus on linear and exponential functions ex solve the following equations by using a table give your answer to the nearest tenth

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p. 13

a-rei.12 graph the solutions to a linear inequality in two variables as a half plane excluding the boundary in the case of a strict inequality and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes a-rei.12 understand that all points on a half-plane are solutions to a linear inequality ex how do we use a graph to represent the solutions to a linear inequality why do we use a graph instead of listing the solutions as we do when solving equations a-rei.12 determine whether the boundary line should be included as part of the solution set ex decide whether the boundary line should be included for the following inequalities how many solutions does each inequality have a-rei.12 graph the solutions to a linear inequality in two variables as a half-plane excluding the boundary for non-inclusive inequalities ex graph the following inequalities a-rei.12 understand that the solutions to a system of inequalities in two-variables are the points that lie in the intersection of the corresponding half-planes ex compare the solution to a system of equations to the solution of a system of inequalities ex describe the solution set of a system of inequalities a-rei.12 graph the solution set to a system of linear inequalities in two variables as the intersection of their corresponding half-planes ex graph the solution set for the following system of inequalities !

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p. 14

interpreting functions common core cluster understand the concept of a function and use function notation common core standard f-if.1 understand that a function from one set called the domain to another set called the range assigns to each element of the domain exactly one element of the range if f is a function and x is an element of its domain then fx denotes the output of f corresponding to the input x the graph of f is the graph of the equation y fx f-if.2 use function notation evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context unpacking what does this standard mean that a student will know and be able to do f-if f-if.1 the domain of a function is the set of all x-values which you control and therefore is called the independent variable the range of a function is the set of all y values and is dependent on a particular x-value thus called the dependent variable students should experience a variety of types of situations modeled by functions detailed analysis of any particular class of functions should not occur at this level students will apply these concepts throughout their future mathematics courses ex when is an equation a function explain the notation that defines a function ex describe the domain and range of a function and compare the concept of domain and range as it relates to a function f-if.2 using function notation evaluate functions and explain values based on the context in which they are in at this level focus on linear and exponential functions ex evaluate for the function ex the function describes the height h in feet of a tennis ball x seconds after it is shot straight up into the air from a pitching machine evaluate and interpret the meaning of the point in the context of the problem.

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p. 15

f-if.3 recognize that sequences are functions sometimes defined recursively whose domain is a subset of the integers for example the fibonacci sequence is defined recursively by f0 f1 1 fn+1 fn fn-1 for n 1 f-if.3 a sequence can be thought of as a function with the input numbers consisting of the integers and the output numbers being the terms of the sequence connect to arithmetic and geometric sequences f-bf.2 emphasize that arithmetic and geometric sequences are examples of linear and exponential functions in an arithmetic sequence each term is obtained from the previous term by adding the same number each time this number is called the common difference in a geometric sequence each term is obtained from the previous term by multiplying by a constant amount called the common ratio now-next equations are equations that show how to calculate the value of the next term in a sequence from the value of the current term the arithmetic sequence next now ± 0#1 1 1 2 1!3 1 4-13 0-4 1 540 0#1 1 16 2 #1 0 0 7 1 4-13 89#1 4%6 2 7 1 · 1 54-1 1 0 1 1 2 1 4-13 1 540 1 54-1 1 0 1 16 0 2 1 0 0 4-13 89#1 4%6!0 now-next equations are the first step in the process of formalizing a sequence using function notation the now-next representation allows students to explore and understand the concept of a recursive function before being introduced to symbolic notation such as an an-1 6 or fn fn-1 6 in the first course students need only convert between recursive and explicit forms for arithmetic and geometric sequences ex you just got a pair of baby rabbits for your birthday ­ one male and one female you decide that you will breed the rabbits but need to plan a budget for the upcoming year to help prepare your budget you need an estimate of how many rabbits you will have by the end of the year in order to build a mathematical model of this situation you make the following assumptions · · · · · a rabbit will reach sexual maturity after one month the gestation period of a rabbit is one month once a female rabbit reaches sexual maturity she will give birth every month a female rabbit will always give birth to one male rabbit and one female rabbit rabbits never die so how many male/female rabbit pairs are there after one year 12 months ex in august 2011 the population in the united states was approximately 312 million suppose that in recent trends the birth rate was 1.7 of the total population use the word now to represent the population of the united

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