Big Idea(s): Algebra is distinguished from arithmetic by the systematic use of symbols for values. Writing and evaluating expressions with algebraic notation follows the same rules/properties as in arithmetic. These expressions and equations can be used to model real world situations. Essential Question Set #1 How are algebraic expressions simplified? How do the basic properties help us understand simplifying expressions? Concepts/Skills/ Commutative and Associative Properties Distributive property Writing expressions How does math help us describe and understand the real world? Concepts/Skills Drawing a diagram Using Formulas Alg. : 1.0, 2.0(PWR), 4.0(PWR), 5.0(PWR), 10.0(PWR), (Refer to Mathematical Reasoning ) Explain why 5x 2 (5x) 2 Give algebraic and numeric examples to support your answer Which property is demonstrated when one combines like terms in an algebraic expression? Give algebraic and numeric examples to support your answer Explain whether the formula for area of a rectangle should be A= l w or A= w l Support your answer with examples What property also supports your assertion? Explain whether finding miles per hour or hours per mile gives you the same result. Support your answer with examples What does this say about the commutative property? Big Idea(s): The rational numbers include integers and fractions. These numbers can be manipulated, represented on the number line and computed in the manner known to whole numbers Essential Question Set #2 How is the rational number line made? How does computing with all rational numbers differ from just the whole numbers. How is it the same? Concepts/Skills/ Computing with rational numbers (Refer to Mathematical Reasoning ) Given 20 feet of flexible fencing, What is the greatest area that could be enclosed within it using a rectangular shape? Justify your answer with drawings and with numbers. There are pigs and chickens in a yard. When counted, there are 18 animals and 52 feet. How many chickens and how many pigs? Justify your answer with diagrams
Using the distributive property Inverse of a sum Alg. : 1.0, 2.0, 4.0, Algebra of extension Topic: 25.0 EUSD Algebra 1 What is the relationship between the area and perimeter of the different rectangular areas you could enclose using all of the fencing? or drawings. Justify your answer using algebra Write a new but similar problem and provide the algebraic solution. Essential Questions Set #3 How do the addition and multiplication properties of equations help us in mathematics? How do you use the multiplication property for equations to clear decimals or fractions from an equation before solving? Concepts/Skills/ Addition property for equation Multiplication property for equations Proportions Using Percents Alg. : 5.0, 24.0, 25.0 Explain how multiplying 3 2 1 is similar and how it is 2 different than multiplying a(b + c). Use any appropriate math vocabulary you have learned to support your claim. (a b) = 12. Give two examples of values for a and b that would make this equation true. True or false, if (a b) = 12, then (a b) = 12 Support your answer with algebra and with numerical examples Test: Chapters 1 and 2 Big Idea(s): Solving and simplifying to solve equations is performed using the addition and multiplication properties of equality in addition to the field properties. Big Ideas and Essential Question Set #4 Why and how does the graph of an inequality differ from that of an equality? How do the addition and multiplication properties of inequalities help us in mathematics? Describe the type of real world problems that are represented by inequalities. (Refer to Mathematical Reasoning )
/ graph inequalities solve inequalities with one variable write inequalities to solve real life problems use logical reasoning to solve problems Alg. : 1.0, 5.0, 24.0 Big Idea(s): Numbers can be written in exponential notation. Multiplying with numbers in exponential notation require the use of exponent rules. (Refer to Mathematical Reasoning ) Essential Question Set #5 Why does 3x 2 (3x) 2? Why does (x 2 ) 3 = x 6 and not x 8? Why do we use scientific notation? / multiply, divide, and raise a power to a power using exponents use scientific notation Alg. : 2.0, 5.0, 10.0 Big Idea(s): Polynomials are algebraic expressions with one (monomials) or more terms. These terms are separated by addition when expressed. Operations on polynomials require combining like terms (addition) and the distributive property (multiplication and factoring). Solving polynomial equations utilize the same properties as in linear equations. Solving polynomial equations is easier when you recognize certain special products. Essential Question Set #6 What does the term polynomial mean? How is a trinomial such as x 2 + 3x + 1 similar to a 3 digit number? What is a perfect square trinomial? What is the difference of two squares? add, subtract, and multiply polynomials recognize special products of binomials Alg. : 10.0(PWR) (Refer to Mathematical Reasoning ) What would 3n 2 + 4n + 1 be equal to if: n = 10? n = 3? How could I add or subtract from x 2 + 6x + 16 to make a perfect square trinomial. Give as many examples as you can think of.
Essential Question Set #7 How is a trinomial square different than a difference of to squares? How does an area model demonstrate the product of 2 polynomials. How does the zero product principal help you solve equations? How does factoring help you find solutions to a polynomial equation? factor special products binomial and trinomial squares use general strategies to factor and solve polynomials solve polynomial equations by factoring Alg. : 10.0(PWR), 11.0, (PWR) 14.0(PWR) Benchmark 1 Algebra : 1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable. 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. 3.0 Students solve equations and inequalities involving absolute values. 4.0 Students simplify expressions prior to solving linear equations and inequalities in one variable, such as 3(2x 5) + 4(x 2) = 12. 5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. 11.0 Students apply basic factoring techniques to second and simple third degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials. 25.0 Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements.
Big Idea(s): Linear functions can be represented as lines on the Cartesian plane. The equation of a line tells you the slope and y intercept of a line as well as the relationship between the x and y coordinates for any point on that line. Essential Question Set #8 Slope is a rate. How is that rate represented when a line is graphed? How can you find the equation of a line just by looking at its graph? What is the difference in the equations of a line and a line perpendicular to that line? parallel to that line? graph linear equations with two variables write an equation of a line find an equation of a line that models given data Alg. : 6.0(PWR); 7.0(PWR); 8.0, (Refer to Mathematical Reasoning ) Big Idea(s): Two or more linear functions considered together are called a system. Solving a system involves methods of substitution, addition and properties of equations or graphing. Many common applied problems are represented by systems of equations. Essential Question Set #9 How is the solution to a system of equations represented on a graph? When one use the substitution or addition method for solving a system of equations, what is the first goal of these methods? How are digit and coin problems similar and different than motion problems? solve systems of two equations with two variables by graphing solve systems of equations by using the substitution and (Refer to Mathematical Reasoning )
addition methods Applying system solving techniques to applied problems such as motion and digit/coin problems. EUSD Algebra 1 Alg. : 9.0(PWR), 15.0(PWR) Big Idea(s): The union and intersection of sets can be described algebraically and geometrically. Set builder notation facilitates the algebraic representation. Absolute value equations may represent a disjunction or a conjunction in a solution. The solution of algebraic inequalities can also be a conjunction or a disjunction. (Refer to Mathematical Reasoning ) Essential Question Set #10 How is set builder notation different than roster notation an why would you use each? What does it mean when the solution set for a compound sentence is a conjunction? A disjunction? Identify unions and intersections of sets Compound Sentences Alg. : 1.0; 5.0(PWR); 24.0 Big Idea(s): Equations in absolute value and inequalities differ from the equations we have learned before by having more than one solution. Solutions of absolute value equations are disjunctions and can have 1 or 2 solutions. Solutions of inequalities are a set of solutions greater than 1. Solutions to an inequality in two variables is a ½ plane in the Cartesian coordinate plane with the boundary the graph of the inequality considered as an equation. Essential Question Set #11 How is the solution to an absolute value equation found? How is the solution to an inequality found? What does the graph of a disjunction look like? What does the graph of the solution of an inequality on two variables look like? (Refer to Mathematical Reasoning )
Solving and graphing absolute value equations. Solving and graphing inequalities in one variable. Solving and graphing inequalities in two variables. Graphing systems of linear inequalities Alg. : 3.0(PWR); 6.0(PWR); 9.0(PWR) EUSD Algebra 1 Big Idea(s): Rational expressions are the quotient of two polynomials. Computation with rational expressions generalizes the methods used for working with fractions (arithmetic rational expressions) utilizing algebra skills. Essential Question Set #12 How are rational expressions simplified? How are rational expressions combined? How are rational expressions multiplied and divided? In general, how do we solve rational equations? Simplifying rational expressions Computing with rational expressions Solving rational equations Applying rational equations Alg. : 11.0 (PWR); 12.0(PWR); 13.0(PWR) (Refer to Mathematical Reasoning ) Big Idea(s): Rational expressions and equations can be used to solve various rate problems. In work problems, the rate of work is a ratio. Mixture problems apply percent to systems of equations in two variables. In those involving liquid, for example, the two equations represent the total solution and the total amount of the non water ingredient in the solution. Essential Question Set #13 Why are rational expressions used to solve work problems? (Refer to Mathematical Reasoning )
What do the two equation in the method for solving mixture problems usually represent? Solving rate problems Solving mixture problems. EUSD Algebra 1 Algebra : 15.0(PWR) Benchmark 2 Algebra : 1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable. 3.0 Students solve equations and inequalities involving absolute values. 5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. 6.0* Students graph a linear equation and compute the x and y intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4). 7.0* Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations using the point slope formula. 8.0 Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point. 9.0* Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. 11.0 Students apply basic factoring techniques to second and simple third degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials. 12.0* Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms. 13.0* Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. 15.0* Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.
Extension Only Big Idea(s): Division of polynomials by a monomial utilizes factorization. Division of a polynomial by a polynomial may require long division. Remainders may be rational expressions. Complex rational expressions are simplified using the LCM. Essential Question Set #14 What is the method for dividing polynomials? What is a complex rational number? How are complex rational numbers simplified? Dividing rational numbers Simplifying complex rational numbers. Alg. : 12.0 (PWR); 13.0(PWR) (Refer to Mathematical Reasoning ) Big Idea(s): The real number set includes rational and irrational numbers. Irrational numbers include the square root of non perfect squares. Algebraic irrationals are written as radical expressions, utilizing the radical symbol. Radical expressions can be simplified using factorization. Multiplication of radical expressions involves the rules of exponents and the careful application of the distributive property. Addition of radical expressions involves using the distributive property. Essential Question Set #15 What is a radical expression? What is the method for simplifying radical expressions? What are the rules to follow for multiplying radical expressions? Simplifying radical expressions. Multiplying radical expressions Adding & subtracting radical expressions Alg. : 2.0(PWR) (Refer to Mathematical Reasoning )
Big Idea(s): The Pythagorean theorem states that, for a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse. The Pythagorean theorem can be used to solve for side lengths of right triangles. Radical equations are solved by squaring both sides to free the unknown from inside the radical. Essential Question Set #16 What relationship does the Pythagorean Theorem describe? How do we use the Pythagorean theorem to find unstated dimensions of a triangle? The Pythagorean Theorem Using the Pythagorean Theorem The principle of squaring Alg. : 2.0(PWR) Taking a root (Refer to Mathematical Reasoning ) Big Idea(s): An equation that is written in the form ax 2 +bx+c=0, where a, b, c are real numbers and a 0, is called a quadratic equation. Quadratics can be solved by factoring, completing the square, or by using the quadratic formula. Essential Question Set #17 What is a quadratic equation? How do you solve quadratic equations? Why do we complete the square? What does the quadratic formula find? Quadratic equations Solving quadratic equations Completing the square The quadratic formula Alg. : 13.0; 14.0(PWR); 19.0; 20.0(PWR); 21.0; 22.0; 23.0(PWR) (Refer to Mathematical Reasoning )
Big Idea(s): A functional is a relation that assigns to each member of the domain exactly one member of the range. The members of the domain can be called inputs and the range the outputs. For a given function output values can be found by evaluating for specific input values. Relations and functions can be graphed on the Cartesian coordinate plane. Functions of the form y=mx+b are called Linear Functions. Functions of the form y=ax 2 +bx+c are called Quadratic Functions. (Refer to Mathematical Reasoning ) Essential Question Set #18 What makes a relation a function? Can you describe a linear function algebraically and graphically? Can you describe a quadratic function algebraically and graphically? Functions Domain and Range Linear functions Quadratic functions Algebra : 16.0; 17.0; 18.0 Big Idea(s): An equation of the form y=kx, where k is a constant expresses a direct variation. Equation of the form y =k/x, where k is a constant expresses inverse variation. An equation of the form z=kxy, expresses joint variation. Essential Question Set #19 What are examples of direct variation? What are examples of inverse variation? What are examples of joint variation? What are examples of combined variation? Direct Variation Inverse Variation Combined and Joint Variation Algebra : 15.0(PWR) Equation of the form z =kx/y, where k is a constant expresses combined variation. (Refer to Mathematical Reasoning )
Benchmark 3 Algebra : 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. 12.0* Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms. 13.0* Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. 14.0* Students solve a quadratic equation by factoring or completing the square. 15.0* Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. 16.0 Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions. 17.0 Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression. 18.0 Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion. 19.0* Students know the quadratic formula and are familiar with its proof by completing the square. 20.0* Students use the quadratic formula to find the roots of a second degree polynomial and to solve quadratic equations. 22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x axis in zero, one, or two points. 21.0* Students graph quadratic functions and know that their roots are the x intercepts. 23.0* Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.
Extended Topics Application of Skills: Problems involving rates such as motion problems are the most direct application of algebra students will experience in High School outside of mathematics courses (Physics, Economics, etc). For first year Algebra students, the focus is on problem solving strategies and communication of method and solution through algebra. Essential Question Set How is algebra used to solve applied problems? How are systems of equations and inequalities used to solve various types of applied problems? In what ways do models, such as drawings or physical models help us in problem solving or understanding a concept How does a solution in algebra translate into a graphic representation? Applying algebra to word problems Accessing notes and other text based examples for use in problem solving Using models to solve problems Using graphs to explain a solution or solve a problem Proofs and Reasoning Essential Question Assessment Samples Big Idea(s): Mathematical and algebraic statements can be proven using valid arguments and proofs Essential Question Assessment Samples What is a proof? What is a reasoning Strategy? Writing, reading a two column proof Working backwards to solve a word problem Alg. : 24.0; 25.0; 25.1 Essential Question Set
Growth and Decay Big Idea(s): Applications and projects involving growth and decay apply functional analysis to life science and social science topics such a s bacterial growth and population growth. Essential Question Set What is the difference between linear and exponential growth? Demonstrate linear growth algebraically, with models and graphically. Demonstrate exponential growth algebraically, with models and graphically. Analyzing patterns algebraically Graphing functions Modeling a function Comparing linear to nonlinear functions graphically Alg. : 2.0(PWR); 15.0(PWR); 16.0 (Refer to Mathematical Reasoning )