Algebra 1 Unit 3: Systems of Equations s5

1 Enduring understanding (Big Idea): Students will understand that manipulation of polynomials will enable them to model and analyze real-world non-linear situations.

Essential Questions: 1. How are the properties of real numbers related to polynomials? 2. How can two algebraic expressions that appear to be different be the same? 3. What are the characteristics of a quadratic function? 4. How can you use functions to model real-world situations? Students will know… Students will be able to… A.CED.1: I can write equations in one variable and apply  b  b2  4ac  The Quadratic Formula x  them to the real world. I can write inequalities in one variable BY THE END OF THIS UNIT: 2a and apply them to the real world. A.CED.2: I can write/create an equation with 2 or more  Rules of exponents – (focus on negative, zero variables. and fractional exponents I can create a coordinate plane using appropriate labels and scales.  I can graph an equation on a coordinate plane with 2 or more Types of Factoring (excluding sum or variables. difference of cubes) I can represent/ interpret/ identify relationships between quantities from equations and graphs.  Special products A.CED.3: I can represent constraints by equations or inequalities. I can represent constraints by systems of Vocabulary: Greater than, less than, at most, at equations/inequalities. least, =, < ,>, ≤ ,≥ , no more than, no less than, I can interpret solutions as viable or nonviable solutions graphically. Variable, dependent variable, independent variable, A.CED.4: I can rearrange/rewrite formulas to solve for a domain, range, scale, constant, variable, formula, given variable, using the same steps to solve equations. literal equation, exponents, factors, terms, bases, A.SSE.1a: I can identify the parts of an expression, including, its terms, its factors, and its coefficients. coefficients, expression, polynomial, trinomial, I can interpret the parts of an expression, including, its terms, binomial, linear terms, quadratic terms, scale, units its factors, and its coefficients. A.SSE.1b: I can interpret a complex expression by dissecting of measurement, units rates, modeling, quantity, unit it into individual parts. A.SSE.2: I can recognize the patterns of special expressions Unit Resources (i.e. difference of perfect squares or sum of perfect cubes) I Learning Task: can rewrite special expressions. Concept Byte: Using Models to Factor A.SSE.3a: I can factor a quadratic expression. Performance Task: I can identify (reveal) the zeros of a quadratic expression. Ch 8 of Alg. 1 textbook pg 67 – “Pull-it-all-together” A.SSE.3b: I can complete the square of a quadratic expression. I can find the maximum or minimum value of a tasks #1, 2, and/or 3 quadratic function. A.SSE.3c: I can rewrite exponential expressions by using Ch 9 of Alg. 1 textbook pg 151 – “Pull-it-all- properties of exponents. I can rewrite exponential functions by using properties of together” task #1, 2, CCSS Included: exponents. A.APR.1: I can add polynomials. I can subtract polynomials. A.CED.1, A.CED.2, A.CED.3, A.CED.4, A.SSE.1, I can multiply polynomials A.SSE.2, A.SSE.3, A.APR.1, A.REI.1, A.REI.3, A.REI.1: I can explain each step in solving equations using properties of equality. I can give (provide) a reasonable A.REI.4, N.Q.1, N.Q.2, N.Q.3, N.RN.1, N.RN.2 (viable) explanation for each step. A.REI.3: I can solve equations with one variable, including Released Test Questions: equations with coefficients. I can solve inequalities with one variable, including inequalities with coefficients. 3, 5, 8, 13, 16, 20, 33, 38 A.REI.4a: I can solve a quadratic equation by completing the square. I can derive the quadratic formula by completing the square. I can write a quadratic equation as a binomial square. A.REI.4b: I can solve quadratic equations using square roots. I can solve quadratic equations by completing the square. I can solve quadratic equations by factoring. I can solve quadratic equations by inspection. I can identify which method to use to solve a quadratic equation. N.Q.1: I can identify the units in problems. I can include the Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. 2 CORE CONTENT Cluster Title: Create equations that describe numbers or relationships Standard: 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. Concepts and Skills to Master:  Create one-variable linear equations and inequalities from contextual situations (stories).  Create one-variable exponential equations and inequalities from contextual situations (stories).  Solve and interpret the solution to multi-step linear equations and inequalities in context.  Use properties of exponents to solve and interpret the solution to exponential equations and inequalities in context. SUPPORTS FOR TEACHERS Critical Background Knowledge  Understand and use inverse operations to isolate variables and solve equations.  Efficiently use order of operations  Understand notation for inequalities  Understand and use properties of exponents Academic Vocabulary Greater than, less than, at most, at least, =, < ,>, ≤ ,≥ , no more than, no less than Suggested Instructional Strategies: Resources:  Convert contextual information into  Textbook Correlation: 1-8, 2-1, 2-2, 2-3, 2-4, mathematical notation. 2-5, 2-7, 2-8, 3-2, 3-3, 3-4, 3-6, 3-7, 7-1, 7-3,  Use story contexts to create linear and 7-4, 7-5, 9-3, 9-4, 9-5, 9-6, 11-5 exponential equations and inequalities  MARS Apprentice Tasks: Functions NCDPI Unpacking: Multiplying Cells A-CED.1 From contextual situations, write Printing Tickets equations and inequalities in one variable and  MARS Expert Tasks: use them to solve problems. Include linear and Fearless Frames exponential functions. At this level, focus on Skeleton Tower linear and exponential functions. Best Buy Tickets

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. 3

Skill-based task Problem Task

Juan pays $52.35 a month for his cable bill and Juan pays $52.35 a month for his cable bill and an an additional $1.99 for each streamed movie. If additional $1.99 for each streamed movie. Gail pays his last cable bill was $68.27, how many $40.32 a month for her cable bill and an additional movies did Juan watch? $2.59 for each streamed movie. Who has the better deal? Justify your choice.

Two boys, Shawn and Curtis, went for a walk. Shawn began walking 20 seconds earlier than Curtis.  Shawn walked at a speed of 5 feet per second  Curtis walked at a speed of 6 feet per second

For how many seconds had Shawn been walking at the moment when the two boys had walked exactly the same distance?

CORE CONTENT Cluster Title: Create equations that describe numbers or relationships Standard: A.CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axe with labels and scales. Concepts and Skills to Master:  Write and graph an equation to represent a linear relationship  Write and graph an equation to represent an exponential relationship  Model a data set using an equation  Choose the best form of an equation to model linear and exponential functions SUPPORTS FOR TEACHERS Critical Background Knowledge  Graph points  Choose appropriate scales and label a graph  Understand slope as a rate of change of one quantity in relation to another quantity Academic Vocabulary Variable, dependent variable, independent variable, domain, range, scale Suggested Instructional Strategies: Resources:  Use story contexts to create linear and  Textbook Correlation: 1-9, 4-5, 5-2, exponential graphs. 5-3, 5-4, 5-5, 7-6, 7-7, 9-1, 9-2, 10-5,  Use technology to explore a variety of 11-6, 11-7, CB 11-7 linear and exponential graphs.  MARS Apprentice Tasks:  Use data sets to generate linear and Functions exponential graphs and equations Multiplying Cells Printing Tickets NCDPI Unpacking:  MARS Expert Tasks: A-CED.2 Given a contextual situation, write Fearless Frames equations in two variables that represent the Skeleton Tower relationship that exists Best Buy Tickets 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.

Sample Assessment Tasks Skill-based task Problem Task Write and graph an equation that models the Jeanette can invest $2000 at 3% interest cost of buying and running an air conditioner compounded annually or she can invest $1500 with a purchase price of $250 which costs at 3.2% interest compounded annually. Which is $0.38/hr to run. the better investment and why?

Two times Antonio’s age plus three times Sarah’s age equals 34. Sarah’s age is also five times Antonio’s age. How old is Sarah?

CORE CONTENT Cluster Title: Create equations that describe numbers or relationships Standard: 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 const constraints on combinations of different foods. Concepts and Skills to Master:  Determine whether a point is a solution to an equation or inequality  Determine whether a solution has meaning in a real-world context  Write and use a system of equations and/or inequalities to solve a real-world problem  Recognize that the equations and inequalities represent the constraints of the problem. Use the Objective Equations and the Corner Principle to determine the solution to the problem. (Linear Programming) SUPPORTS FOR TEACHERS Critical Background Knowledge  Ability to read and write inequality symbols  Ability to graph equations and inequalities on the coordinate plane Academic Vocabulary Constraint, greater than, >, less than, <, greater than or equal to, ≥, less than or equal to, ≤ , inequality, viable

Suggested Instructional Strategies: Resources:  Use a story context such as those found  Textbook Correlation: 6-5, 6-6, CC-7 in linear programming problems to write  MARS Concept Development Lessons: and graph equations with constraints. Defining Regions using Inequalities  Analyze real-world problems connected Solving Linear Equations in Two Variables to student interest  MARS Problem Solving Lesson: Optimization problems NCDPI Unpacking: 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.

Sample Assessment Tasks Skill-based task Problem Task Given y ≤ 2x + 1 and y > x – 3 find a point Iced tea costs $1.50 a glass and lemonade costs that: $2.00. If you have $12, what can you buy? Justify your answer using multiple a. Satisfies both representation. b. Satisfies one, but not the other c. Satisfies neither

Paul sells chocolate chip cookies and peanut butter cookies.  Baking a batch of chocolate chip cookies takes 1.75 cups of flour and 2 eggs.

 Baking a batch of peanut butter cookies takes 1.25 cups of flour and 1 egg.

 Paul has 10 cups of flour and 12 eggs.

 He makes $4 profit per batch of chocolate chip cookies

 He makes $2 profit per batch of peanut

 butter cookies

How many batches of peanut butter cookies should Paul make to maximize his profit? A 1 B 2 C 5 D 8

All American Bats, produces two different quality wooden baseball bats, the Aaron Bat and the DiMaggio Bat. The Aaron Bat takes 8 hours to trim and 2 hours to finish it. It has a profit of $16. The DiMaggio Bat takes 5 hours to trim and 5 hours to finish it, but it has a profit of $27. The total time available per day for trimming is 80 hours and 50 hours for finishing. What is the maximum profit the All American Bats can make each day? A $270 B $296 C $366 D $432

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. 9 CORE CONTENT Cluster Title: Create equations that describe numbers or relationships Standard: A.CED.4: Rearrange formulas to highlight a quantity of interest Concepts and Skills to Master:  Extend the concepts used in solving numerical equations to rearranging multi-variable formulas or literal equations to solve for a specific variable. SUPPORTS FOR TEACHERS Critical Background Knowledge  Recognize variables as representing quantities in context  Solve multi-step equations Academic Vocabulary Constant, variable, formula, literal equation Suggested Instructional Strategies: Resources:  Use formulas for a variety of disciplines  Textbook Correlation: 2-5 such as physics, chemistry, or sports to explore the advantages of different formats of the same formula

NCDPI Unpacking: 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 interest or to formulas involving squared or cubed variables.

Sample Assessment Tasks Skill-based task Problem Task

I = Prt Solve for r. Paul just arrived in England and heard the temperature in degrees Celsius. He remembers 5 that C  (F  32) . How will Paul find the 9 Which equation is 5x + 4y = 9x + 8 correctly temperature in Fahrenheit? solved for x? 9x  4y  8 A x  5

B x  3

C x  y  2

5x  8 D x  5

Energy and mass are related by the formula E = mc2  m is the mass of the object  c is the speed of light

Which equation finds m, given E and c?

A m  E  c2 B m  Ec 2

c2 C m  E E D m  c 2

CORE CONTENT Cluster Title: Interpret the Structure of Expressions 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 P(1 + r)n as the product of P and a factor not depending on P. Concepts and Skills to Master:  Given an expression, identify the terms, bases, exponents, coefficients, and factors.  Determine the real world context of the variables in an expression.  Identify the individual factors of a given term within an expression.  Explain the context of different parts of a formula. SUPPORTS FOR TEACHERS Critical Background Knowledge  Understand the meaning of symbols indicating mathematical operations, implied operations (e.g. 2x), the meaning of exponents, and grouping symbols. Academic Vocabulary Exponents, factors, terms, bases, coefficients, expression

Suggested Instructional Strategies: Resources:  Given a word problem and a formula have  Textbook Correlation: 1-1, 1-2, 1-7, students examine the structure and explain 3-7, 4-5, 5-3, 5-4, 7-7, 8-5, 8-6, 8-7, the context of different parts of the formula. 8-8, 9-5, CC-2, CC-10  Design a game around identifying terms, bases, exponents, coefficients, and factors.  MARS Concept Development Lesson:  Create formulas based on context Sorting Equations and Identities NCDPI Unpacking: 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 10,000(1.055)5. 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. A-SSE.1b Students group together parts of an expression to reveal underlying structure. For example, consider the expression 4000p – 250p2 that represents income from a concert where p is the price per ticket. The equivalent factored form, p(4000 – 250p) 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.

Sample Assessment Tasks

Consider the formula Surface Area = 2B + Ph Interpret the expression: 5 – 3(x – y)2. Explain a. What are the terms of this formula? the output values possible. b. What are the coefficients?

Ex. The expression 20(4x) + 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.

CORE CONTENT Cluster Title: Interpret the Structure of Expressions

Standard: A.SSE.2: Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). Concepts and Skills to Master:  Rewrite an algebraic expression in different forms such as factoring or combining like terms  Use factoring techniques such as common factors, grouping, the difference of two squares, the sum or difference of cubes, or a combination of methods to factor completely.  Simplify expressions including combining like terms, using the distributive property and other operations with polynomials. SUPPORTS FOR TEACHERS Critical Background Knowledge  Clear and concrete understanding of the distributive property and factoring polynomials Academic Vocabulary

Suggested Instructional Strategies: Resources:  Use algeblocks or algebra tiles to  Textbook Correlation: 5-3, 5-4, 5-5, explore structure of expressions. 8-7, 8-8, CC-12

NCDPI Unpacking: 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 2(x – 1)2 - 4 to show Interior Design that it is a quadratic expression of the ax2 + bx A square rug has an area of 49x2 -56x + 16. A +c. second square rug has an area of 16x2 + 24x + 9. What is an expression that represents the difference of the areas of the rugs. Show two different ways to find the solution.

Which expression is equivalent to t2 – 36?

A (t – 6)(t – 6) B (t + 6)(t – 6) C (t – 12)(t – 3) D (t – 12)(t + 3)

CORE CONTENT Cluster Title: Write expressions in equivalent forms to solve problems Standard: A.SSE.3: Choose and produce an 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 b. Complete the square in a quadratic expression to reveal the maximum or minimum values of the function it defines. c. Use the properties of exponents to transform expressions for exponential functions. Concepts and Skills to Master:  Given a quadratic function explain the meaning of the zeros of the function. e.g. if f(x) = (x – c)(x – a), then f(a) = 0 and f(c) = 0  Given a quadratic expression, explain the meaning of the zeros graphically. e.g. for an expression (x – c)(x – a), a and c correspond to the x-intercepts (if a and c are real numbers).  Write the vertex form of a quadratic expression by completing the square.  Use the vertex form to find the maximum or minimum of a quadratic function and explain the meaning of the vertex. SUPPORTS FOR TEACHERS Critical Background Knowledge Students should have a good understanding of simple factoring techniques to this point. Students should also understand about the GCF and how to identify or determine the GCF. Academic Vocabulary

Suggested Instructional Strategies: Resources:  Teachers should be familiar with most all  Textbook Correlation: 7-7, CC-15 aspects of trinomial factoring. Teachers should be particularly well versed in factoring using the GCF (greatest common factor). A brief review of factoring with algebra tiles and GCF would greatly help the teacher with this lesson.  Use algeblocks or algebra tiles to explore the meaning of “completing the square” and discover the algorithm NCDPI Unpacking: 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 seconds later. At this level, limit to quadratic expressions of the form ax2 + bx + c.

2 2 The larger leg of a right triangle is 3 cm longer It is given that x + 10x + 24 = (x + 5) + k. than its smaller leg. The hypotenuse is 6 cm Find the value of k. longer than the smaller leg. How many centimeters long is the smaller leg? What is the smallest of 3 consecutive positive integers if the product of the smaller two integers is 5 less than 5 times the largest integer?

CORE CONTENT Cluster Title: Perform Arithmetic Operations on Polynomials Standard: A.APR.1: Understand that 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. Concepts and Skills to Master:  Look for and make use of the structure of addition and subtraction extended to linear and quadratic polynomials  Understand the concept of like terms and closure  Identify and make use of the structure of special products of linear binomial expressions. SUPPORTS FOR TEACHERS Critical Background Knowledge  Distributive Property  Addition, subtraction and multiplication of integers Academic Vocabulary Polynomial, trinomial, binomial, linear terms, quadratic terms

Suggested Instructional Strategies: Resources:  Textbook Correlation: 8-1, 8-2, 8-3, 8-4 NCDPI Unpacking:  MARS Concept Development Lesson: Interpreting Algebraic Expressions 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 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.

2 Which expression is equivalent to r + 2(rw – The area of a trapezoid is found using the 2 x) when r = -3 and x = -1? 1 formula A  h(b1  b2 ) , where A is the area, h is 2 2 A 18w + 12 w + 11 the height, and b1 and b2 are the lengths of the 2 B 18w – 12w + 11 bases. 2 C 99w + 66w + 11 2 D 99w – 66w + 11

What is the area of the above trapezoid?

A A = 4x + 2 B A = 4x + 8 2 C A = 2x + 4x – 21 2 D A = 2x + 8x – 42

CORE CONTENT Cluster Title: Reason quantitatively and use units to solve problems 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. Concepts and Skills to Master:  Select and use appropriate units of measurement for problems with and without context  Given a graph draw conclusions and make inferences  Choose appropriate scales to create linear and exponential graphs  Determine from the labels on a graph what the units of the rate of change are (e.g. gallons per hour) SUPPORTS FOR TEACHERS Critical Background Knowledge  Know how various attributes are reasonably measured Academic Vocabulary Scale, units of measurement Suggested Instructional Strategies: Resources:  Explore a variety of examples of measurements  Textbook Correlation: 2-5, used in graphs CB 2-5, 2-6, 2-7, 4-4, 5-7,  Construct graphs using a variety of data sets 12-2, 12-4

NCDPI Unpacking: 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: (mi/hr)(hr) = mi (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. 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.

Sample Assessment Tasks Skill-based task Problem Task What is the area of strip of wall that is 48 inches by 10 Your college savings fund has $1800 yards? in it and you plan to spend $30 a week. What would be an appropriate viewing window and scale to see the remaining balance each week until the money is gone? Explain.

A radio-controlled car traveled 30 feet across the classroom in 1.6 seconds. Ex. When finding the area of a circle Use appropriate quantities from the box to determine using the formula ! ! !!!, which unit of how fast the car traveled in miles per hour. measure would be appropriate for ______x______x______x______the radius? a. square feet = b. inches 3 ft 5280 ft 1yd 1mi 12in c. cubic yards d. pounds 1yd 1mi 3 ft 5280 ft 1 ft

1ft 60 min 1hr 60 sec 1min

Standards are12 listedin in alphabetical1hr 60 /numericalmin 1 ordermin not60 suggestedsec teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. 24

Ex. Based on your answer to the previous question, what units would the area be measured in?

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?

Alex walked 1 mile in 15 minutes. Sally walked 3520 10, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 23, 24, 25, 27, 31, 34, 34, 35, 35, yards in 24 minutes. In miles per hour, how much faster 36, 37, 38, 39, 39, 41, 43, 44, 45, 48, did Sally walk than Alex? (Note: 1 mile = 1760 yards) 49, 62, 73

CORE CONTENT Cluster Title: Reason quantitatively and use units to solve problems Standard: N.Q.2: Define appropriate quantities for the purpose of descriptive modeling Concepts and Skills to Master:  Choose appropriate measures and units for problem situations  Create a relationship among different units (i.e. feet per second, bacteria per hour, miles per gallon) SUPPORTS FOR TEACHERS Critical Background Knowledge  Compute unit rates associated with ratios of fractions  Recognize and calculate basic conversions (e.g. 3 feet = 1 yard) Academic Vocabulary

Units rates, modeling, quantity, unit conversion, proportion, ratio Suggested Instructional Strategies: Resources:  Integrate this objective into problem  Textbook Correlation: 2-6, 3-3, 4-5, solving throughout the curriculum. 5-2, 5-5, 12-3, CC-7  Place an emphasis on relationships between two different units (e.g. dollars per hour, pressure over altitude, calories per gram)

NCDPI Unpacking:

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.

Skill-based task Problem Task How would you measure the rate at which a Recreate a scenario you have encountered bathtub fills? Justify your answer. involving two changing quantities and determine appropriate units to describe the relationship between the quantities. 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?

CORE CONTENT Cluster Title: Reason quantitatively and use units to solve problems

Standard: N.Q.3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities Concepts and Skills to Master:  Determine whether whole numbers, fractions, or decimals are most appropriate  Determine the appropriate power of ten to reasonably measure a quantity  Determine the resulting accuracy in calculations  Determine what level of rounding should be used in a problem situation SUPPORTS FOR TEACHERS Critical Background Knowledge  Understand basic units of measurements and their relationship to one another (e.g. a foot is smaller than a yard)  Understand how to properly round and estimate Academic Vocabulary Precision, accuracy Suggested Instructional Strategies: Resources:  Discuss misconceptions in resulting  Textbook Correlation: 2-10, 9-5, 9-6, calculations involving measurement, e.g. CC-7 you cannot increase accuracy through calculation, only through more accurate  MARS Apprentice Task (N.Q.1 through 3): measurement Leaky Faucet  Compare the difference between rounding at different places in a  MARS Apprentice Task (N.Q.1 through 3): calculation and discuss which yields the Yogurt best result  Discuss how mathematicians maintain precision through the representations that they use.

NCDPI Unpacking:

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?

CORE CONTENT Cluster Title: Extend the properties of exponents to rational exponents 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 51/3 to be the cube root of 5 because we want (51/3)3 = 51/3*3 to hold so (51/3)3must equal 5 Concepts and Skills to Master:  Understand that the properties of integer exponents extend to rational exponents. SUPPORTS FOR TEACHERS Critical Background Knowledge  Properties of integer exponents Academic Vocabulary Exponent, rational, radical Suggested Instructional Strategies: Resources:  Students can investigate by considering  Textbook Correlation: CC-8 patterns such as (34) = 31 and ( 4 34 ) = 31 to arrive at a conjecture.

NCDPI Unpacking: N-RN.1 In order to understand the meaning of rational exponents, students can initially investigate them by considering a pattern such as:

CORE CONTENT Cluster Title: Extend the properties of exponents to rational exponents Standard: N.RN.2: Rewrite expressions involving radicals and rational exponents using the properties of exponents. Concepts and Skills to Master:  Rewrite numbers with rational exponents in radical form.  Rewrite numbers in radical form as number with rational exponents. SUPPORTS FOR TEACHERS Critical Background Knowledge  Properties of integer exponents Academic Vocabulary Exponent, rational, radical Suggested Instructional Strategies: Resources:  Textbook Correlation: CC-9 NCDPI Unpacking:

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.

Which expression is equivalent to 3 8x2 y3 z 4 ?

3 3 A 2x 2 yz 4 2 4 B 2x 3 yz 3 2z C x 2x D z

1 3 Which expression is equivalent to x 3  ?

A x 1 B x

1 C x9 1 D x 27

CORE CONTENT Cluster Title: Understand solving equations as a process of reasoning and explain the reasoning Standard: 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. Concepts and Skills to Master:  Understand, apply, and explain the results of using inverse operations  Justify the steps in solving equations by applying and explaining the properties of equality, inverse and identity. Justifications may include the associative, commutative, and division properties, combining like terms, multiplication by 1, etc.  Use the names of the properties and common sense explanations to explain the steps in solving an equation SUPPORTS FOR TEACHERS Critical Background Knowledge  Use order of operations  Simplify expressions using properties of algebra Academic Vocabulary Constant, coefficient, properties of operations and properties of equality, like terms, variable, evaluate, justify, viable

Suggested Instructional Strategies: Resources:  Have students share different ways of  Textbook Correlation: 2-2, 2-3, 2-4, solving equations that lead to the same 2-5, 9-4, 9-5 answer.  Find and analyze mistakes in student work samples  Partner problems: one student solves, the other writes reasons why steps work.  Introduce a two-column proof as a way of organizing justifications

NCDPI Unpacking:

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.

Skill-based task Problem Task Justify the equation solution by writing the When Sally picks any number between 1 and property or reason why each step works. 20, doubles it, adds 6, divides by 2 and 3x + 7 = 12 subtracts 3, she always gets the number she 3x + 7 – 7 = 12 – 7 started with. Why? Evaluate and use algebraic 3x + 0 = 5 evidence to support your conclusion. 3x = 5 (3x)(1/3) = (5)(1/3) 1x = 5/3 x = 5/3

MULTIPLE CHOICE: Dave correctly solved the equation below using 4 steps.

Given: 5(3x + 2) – 7x = 34

Step 1: 15x + 10 – 7x = 34

Step 2: 8x + 10 = 34

Step 3: 8x = 24

Step 4: x = 3

Between which 2 steps did Dave use the Subtraction Property of Equality?

A Between Given and Step 1 B Between Steps 1 and 2 C Between Steps 2 and 3 D Between Steps 3 and 4

Ex. Solve 5(x+3)-3x=55 for x. Use mathematical properties to justify each step in the process.

CORE CONTENT Cluster Title: Solve equations and inequalities in one variable Standard: A.REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Concepts and Skills to Master:  Write equations in equivalent forms to solve problems.  Analyze and solve literal equations for a specified variable.  Understand and apply the properties of inequalities.  Verify that a given number or variable is a solution to the equation or inequality.  Interpret the solution of an inequality in real terms. SUPPORTS FOR TEACHERS Critical Background Knowledge  Solving linear equations Academic Vocabulary Properties of inequality as interpreted in table 5 of the CCSS glossary page 90 Suggested Instructional Strategies: Resources:  Solve specified variables, using common  Textbook Correlation: 2-1, 2-2, 2-3, formulas used in science, economics, or 2-4, 2-5, 2-7, 2-8, 3-2, 3-3, 3-4, 3-5, other disciplines 3-6  Examine and prove why dividing or multiplying by a negative reverses the inequality sign.  Use applications from a variety of disciplines to motivate solving linear equations and inequalities

NCDPI Unpacking:

A-REI.3 Solve linear equations in one variable, including coefficients represented by letters. A-REI.3 Solve linear inequalities in one variable, including coefficients represented by letters.

Sample Assessment Tasks Skill-based task Problem Task The perimeter of a rectangle is given by P = 2W Solve 2(x + 4) – 3 ≥ 4x – 2 + 2L. Solve for W and restate in words the meaning of this new formula in terms of the meaning of the other variables.

A school purchases boxes of candy bars. Ex. Grandma’s house is 20 miles away and  Each box contains 50 candy bars Johnny wants to know how long it will take to  Each box costs $30 get there using various modes of transportation.

How much does the school have to charge for a. Model this situation with an equation where time is a function of rate in miles per hour. each candy bar to make a profit of $10 per b. For each mode of transportation listed below, box? determine the time it would take to get to Grandma’s. A $0.40 Mode of Transportation B $0.50 Rate of Travel in mph C $0.80 Time of Travel hrs. D $1.25 bike 12mph

car 55mph Ex. Solve, Ax +B =C for x. What are the walking 4mph specific restrictions on A?

Ex. What is the difference between solving an Ex. A parking garage charges $1 for the first equation and simplifying an expression? 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.

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. 39 CORE CONTENT Cluster Title: Solve equations and inequalities in one variable Standard: A.REI.4: a. Use the method of completing the square where the leading coefficient of x2 is one (ax2 + bx + c where a=1) to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Concepts and Skills to Master:  Use the method of completing the square where the leading coefficient of x2 is one (ax2 + bx + c where a=1) to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions.  Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. SUPPORTS FOR TEACHERS Critical Background Knowledge  Square Root  Perfect Square Trinomials  Factoring Quadratic Equations  Zeros, Roots, Solutions Academic Vocabulary Properties of inequality as interpreted in table 5 of the CCSS glossary page 90 Suggested Instructional Strategies: Resources: Use Algebra Tiles to develop the understanding  Textbook Correlation: 9-5 of what is meant by “completing the square”  Using Algebra tiles to complete the square  Video of Completing the Square Completing the square lesson plan

Solve the following by completing the square

x2 + 6x – 7 = 0

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.