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 |
| |
|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. |
| | |
| | |
|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 | |
| | |
|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 | |
| | |
| | |
| | |
| | |
|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: |
| |
|NCDPI Resources: |
| |
| |
|National Library of Manipulatives |
| |
|NCTM Illuminations |
| |
|Lesson Plan sites and Activities: |
| |
| |
|Math Graphic Organizers |
| |
| |
|Problem Solving/Problem Websites |
| |
| |
| |
| |
| |
|Currituck County Schools – Common Core Resources |
| |
| |
|AVID Library/Mathematics Write Path I and II |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related searches
- algebra 1 worksheets 7th grade
- algebra 1 worksheets and answers
- algebra 1 worksheets answer keys
- algebra 1 worksheets free printable
- algebra 1 lesson 3 8
- free algebra 1 worksheets with answers
- mcdougal algebra 1 answer key
- algebra 1 worksheets with answers
- math algebra 1 worksheets
- mcgraw hill algebra 1 pdf
- algebra 1 mcgraw hill book
- algebra 1 mcgraw hill answers