Subject: Algebra 1



|Subject: 8th/CC Math I |Timeframe Needed for Completion: 9 weeks |

|Grade Level: 8th | |

|Unit Title: Systems and polynomial functions |Grading Period: 3rd nine weeks |

|Big Idea/Theme: Solving systems of equations and inequalities, understanding and examining polynomial functions |

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|Understandings: |

|Interpret the structure of expressions |

|Write expressions in equivalent forms to solve problems |

|Perform arithmetic operations on polynomials |

|Create equations that describe numbers or relationships |

|Solve equations and inequalities in one variable |

|Solve systems of equations |

|Understand solving equations as a process of reasoning and explain the reasoning |

|Represent and solve equations and inequalities graphically |

|Extend the properties of exponents to rational exponents |

|Analyze functions using different representations |

|Build a function that models a relationship between two quantities |

|Build new functions from existing functions |

|Essential Questions: |Curriculum Goals/Objectives (to be assessed at the end of the unit/quarter) |

|What can equations tell us about everyday circumstances we deal with in life? |1. A.CED.2: Create equations in two or more variables to |

|Can we examine many relationships in our everyday life and develop equations and graphs? |represent relationships between quantities; graph |

|How can solving equations help us develop arguments and justifications? |equations on coordinate axes with labels and scales. |

|Given equations for related quantities, how do we determine the solution? Does every pair of equations |2. A.CED.3: Represent constraints by equations or inequalities, and |

|have a solution? |by systems of equations and/or inequalities, and |

|What does the graph of an equation in two variables provide us? |interpret solutions as viable or non-viable options in a |

|Why does the graph of an inequality in two variables have areas which are shaded? |modeling context. For example, represent inequalities |

|In a graph of an inequality in two variables how do we determine if a solution is on the boundary |describing nutritional and cost constraints on |

|What operations do we use when multiplying and dividing exponents? |combinations of different foods. |

|Can a number have roots other than square roots? |3. A.REI.1: Understand solving equations as a process of reasoning |

|How are polynomial expressions used in real life. |and explain the reasoning |

|What operations can, and cannot be performed with polynomials |4. A.REI.5 Prove that given a system of two equations in two |

|Can expressions be rewritten to always be dependent on a specific entity? |variables, replacing one equation by the sum of that |

|Describe different methods which can be used to produce equivalent forms of an expression |equation and a multiple of the other produces a system |

|How does a variable change how we solve equations and inequalities? |with the same solutions. |

|How does graphing a function help us understand the function? |5. A.REI.6 Solve systems of linear equations exactly and |

|What does comparing various forms of functions allow us to determine? |approximately (e.g with graphs), focusing on pairs of |

|How are arithmetic and geometric sequences functions? |linear equations in two variables. |

|What are even and odd functions? |6. 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) |

| |7. 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. |

| |8. 8.EE.8 Analyze and solve pairs of simultaneous linear equations. |

| |Understand that solutions to a system of two linear |

| |equations in two variables correspond to points of |

| |intersection of their graphs, because points of intersection |

| |satisfy both equations simultaneously. Solve systems of |

| |two linear equations in two variables algebraically, and |

| |estimate solutions by graphing the equations. Solve |

| |simple cases by inspection. For example 3x + 2y = 5 and |

| |3x +2y =6 have no solution because 3x+2y can’t |

| |simultaneously be both 5 and 6. Solve real world and |

| |mathematical problems leading to two linear equations in |

| |two variables. For example, given coordinates for two |

| |pairs of points, determine whether the line through the |

| |first pair of points intersects the line through the second |

| |pair. |

| |9. 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) 3 must equal 5. |

| |10. N.RN.2 Rewrite expressions involving radicals and rational |

| |exponents using the properties of exponents. |

| |11. 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. |

| |12. A.SSE.1 Interpret expressions that represent a quantity in terms |

| |of its context. |

| |Interpret parts of an expression , such as terms, factors |

| |and coefficients |

| |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. |

| |13. 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). |

| |14. A.SSE.3 Choose and produce an equivalent form of an |

| |expression to reveal and explain properties of the |

| |quantity represented by the expression. |

| |Factor a quadratic expression to reveal the zeros of the function it defines. |

| |Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it |

| |defines. |

| |Use the properties of exponents to transform expressions for exponential functions. For example the |

| |expression 1.15t can be rewritten as (1.15 1/12) 12t =1.012 12t to reveal the approximate equivalent |

| |monthly interest rate if the annual rate is 15%. |

| |15. A.REI.3 Solve equations and inequalities in one variable |

| |16. F.IF.7 Graph functions expressed symbolically and show key |

| |features of the graph, by hand in simple cases and using |

| |technology for more complicated cases. |

| |Graph linear and quadratic functions and show intercepts, maxima and minima. |

| |Graph square root, cube root, and piecewise defined |

| |functions , including step functions and absolute value |

| |functions |

| |17. F.IF.8 Write a function defined by an expression in different but |

| |equivalent forms to reveal and explain different properties |

| |of the function. |

| |Use the process of factoring and completing the square |

| |in a quadratic function to show zeros, extreme values, |

| |and symmetry of the graph, and interpret these in terms |

| |of a context. |

| |Use the properties of exponents to interpret expressions |

| |for exponential functions. For example, identify percent |

| |rate of change in functions such a y= (1.02)t, |

| |y = (0.97)t, y = 1.01)12t, y = (1.2) t/10, and classify |

| |them as representing exponential growth or decay. |

| |18. F.IF.9 Compare properties of two functions each represented in a |

| |different way (algebraically, graphically, numerically in |

| |tables or by verbal descriptions. For example, given a |

| |graph of one quadratic function and an algebraic |

| |expression for another, say which has the larger |

| |maximum. |

| |19. F.BF.1 Write a function that describes a relationship between two |

| |quantities. |

| |Determine an explicit expression, a recursive process, or |

| |steps for calculation from a context. |

| |Combine standard function types using arithmetic |

| |operations. For example, build a function that models |

| |the temperature of a cooling body by adding a constant |

| |function to a decaying exponential and relate these |

| |functions to the model. |

| |20. F.BF.3 Identify the effect on the graph of replacing f(x), by f(x) + |

| |k, kf(x), f(kx) and f(x+k) for specific values of k, (both |

| |positive and negative); find the value of k given the |

| |graphs. Experiment with cases and illustrate an |

| |explanation of the effects on the graph using technology. |

| |Include recognizing even and odd functions from their |

| |graphs and algebraic expressions for them. |

| |21. A.REI.4 Solve quadratic equations in one variable. Use the |

| |method of completing the square 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. 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. |

| |22. A.REI.11: Explain why the x-coordinates where the points of the |

| |graphs of the equations y = f(x) and y=g(x) intersect |

| |are the solutions of f(x) = g(x); find the solutions |

| |approximately, (e.g. using technology to graph the |

| |functions), make tables of values or find successive |

| |approximations. Include cases where f(x) and/or g(x0 |

| |are linear, polynomial, rational, absolute value, |

| |exponential and logarithmic functions. |

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|Essential Skills/Vocabulary: |Assessment Tasks: |

|Vocabulary: | |

|x-axis |Quick writes |

|y-axis |Teacher made tests and quizzes |

|system of equations |Find the error |

|boundary |Foldables |

|half-plane |Cornell notes |

|viable |Groupwork |

|non-viable |Projects |

|justify |Graphic organizers |

|elimination |Venn Diagrams |

|substitution |Anticipation/prediction guides |

|rational exponent | |

|radical | |

|monomial | |

|binomial | |

|trinomial | |

|polynomial | |

|terms | |

|factors | |

|coefficients | |

|constants | |

|zeros | |

|completing the square | |

|variables | |

|intercepts | |

|maxima | |

|minima | |

|asymptote | |

|end behavior | |

|recursive process | |

|even function | |

|odd function | |

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|Essential skills: | |

|Create equations with two or more variables to represent relationships between quantities. | |

|Graph equations in two variables on a coordinate plane and label the axes and scales | |

|Write and use systems of equations and/or inequalities to solve real world problems. | |

|Recognize the equations and inequalities represent the constraints of the problem. | |

|Be able to explain each step in solving an equation | |

|Construct arguments to justify a solution method | |

|Solve systems of equations using elimination | |

|Solve systems of equations using substitution | |

|Solve systems of equations using graphing | |

|Understand the definition of the meaning of rational exponents | |

|Extend properties of integer exponents to rational exponents | |

|Rewrite expressions involving radicals and rational exponents | |

|Understand combining like terms and closure | |

|Add, subtract, and multiply polynomials and understand how closure applies under these operations | |

|Identify parts of an expression and explain their meaning in the context of the problem | |

|Rewrite algebraic expressions in different equivalent forms | |

|Write expressions in equivalent forms by factoring and explain the meaning of the zeros. | |

|Write expressions in equivalent forms by completing the square to convey the vertex form | |

|Use properties of exponents to write an equivalent form of an exponential function and explain specific | |

|information about its approximate rate of growth or decay. | |

|Solve one variable equations and inequalities | |

|Graph functions and explain key features; including linear, quadratic, exponential, piecewise, step and | |

|absolute value functions | |

|Write functions defined by expressions | |

|Factor and use completing the square to show key points and interpret in the context of the problem, | |

|quadratic functions | |

|Compare functions represented in different forms | |

|Combine standard function types using arithmetic operations | |

|Compose functions | |

|Identify the effect on graphs when using arithmetic processes. | |

|Identify even and odd functions | |

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|Guiding Questions: | |

|Compare the costs of two different phone plans and determine where each plan is more cost effective. | |

|Develop inequalities describing nutritional and cost constraints on combinations of different foods. | |

|Given an equation, or a problem which can be solved by an equation, construct a convincing argument | |

|justifying each step in the process. | |

|What is the solution of a system of equations? | |

|Describe the possible solutions to a system of equations. | |

|What is the solution of a system of inequalities in two variables? | |

|Describe the possible solutions to a system of inequalities in two variables. | |

|4 1/3 is described as what root of 4? | |

|How do you simplify the square root of 256? | |

|What is necessary to add or subtract polynomials? | |

|What are like terms? | |

|What is closure? | |

|What must you do to multiply polynomials? | |

|In P(1+r) n is the expression dependent on P? | |

|Factor x4 – y4 completely | |

|What do a and c represent in the quadratic expression | |

|(x-a)(x-c)? | |

|Explain, and solve, expressions involving exponents. | |

|What methodology is used to solve simple equations and inequalities? | |

|What is the difference between an equation and an inequality? | |

|What shapes do linear and quadratic graphs have? | |

|How are the domain and range different in linear and quadratic functions? | |

|What is the shape of an absolute value function? | |

|What is the shape of an exponential function? | |

|Which graph do square root and cube root functions resemble? | |

|How do you compare functions represented in different forms? | |

|What is the effect on a graph when we add or subtract a constant to a function? | |

|What is the effect on a graph when we multiply the function? | |

|What is the difference in algebraic expressions for even and odd functions? | |

|Materials Suggestions: |

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|NCDPI Resources: |

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|National Library of Manipulatives |

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|NCTM Illuminations |

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|Lesson Plan sites and Activities: |

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|Math Graphic Organizers |

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|Problem Solving/Problem Websites |

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|Currituck County Schools – Common Core Resources |

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|AVID Library/Mathematics Write Path I and II |

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