# Students will be able to simplify and evaluate numerical and variable expressions using appropriate properties and order of operations.

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1 Outcome 1: (Introduction to Algebra) Skills/Content 1. Simplify numerical expressions: a). Use order of operations b). Use exponents Students will be able to simplify and evaluate numerical and variable expressions using appropriate properties and order of operations. 1. Demonstrate the basic expression 5+3x7. Show the order of operations. Define the rules to simplify the expression accurately. 1. Send students to the board to do sample problems. 2. Understand use of variables: a). Evaluate variable expressions b). Translate word phrases into algebraic expressions. 3. Define and use properties to simplify numerical and algebraic expressions. 4. Evaluate equations using a given replacement set. 2. Demonstrate flashcards with words and corresponding math symbol e.g. > greater than. Have students model (in front of class) a verbal expression and its matching mathematical expression. Then evaluate expressions with a given replacement set. 3. Model commutative and associative properties with beans or similar objects. 4. Students work in pairs to solve given equations with a given replacement set. 2. Have students write a mathematical expression using letters x, y, z and any numbers. Then have them exchange their expressions with another student and evaluate if x=1, y=2 and z=0. 3. Challenge: Have students give examples of why the commutative property does not work for subtraction and division. 4. Give each student a card with a different number on it. Have each student create an equation with that number as a solution.

2 Outcome 2: (Real Numbers) 1. Identify all number sets within the real number system. Students will be able to perform the four basic operations over the real number system. 1. Given a number line, students will locate examples of the different number systems: e.g. integers, whole numbers, natural numbers, rational and irrational numbers. 1. Using a number line drawn on the board, students will identify 5 numbers from the different number sets. 2. Graph and compare real numbers on a number line. 2. After locating a number on the number line, students will place a point and label it with its coordinate. By locating pairs of numbers on the number line, students will discover that the larger number is always to the right. 2. Have students use a number line to answer questions such as: Which numbers are 3 units from the origin? Which numbers are 4 units from the graph of x? Given a set of numbers students will use the number line to graph and order them from least to greatest. 3. Evaluate opposites and absolute values of real numbers. 4. Use number properties to simplify expressions: 3. Use the number line to locate opposites. Relate absolute value to distance from the origin. 4. Given an expression, students will determine the best way to simplify, using the associative and commutative properties. 3. Same as number 2 above. 4. In small groups, students will create a set of problems involving simplifying expressions using the associative and commutative properties.

3 Outcome 2: (Real Numbers) (continued) Students will be able to perform the four basic operations over the real number system. 5. Add and subtract real numbers: a) Additive identity b) Additive inverse 5. Using unit tiles of two different colors students will evaluate sums and differences of real numbers, incorporating additive inverses and additive identity. 5. Using a diagram of a football field, students will record gains and losses to determine net gain/loss. 6. Multiply real numbers: a). Distribute property b). Multiplicative identity c). Multiplicative inverse 6. Using algebra or unit tiles, and the distributive property, students will evaluate products of real numbers. 7. Divide real numbers: a). Reciprocals 7. Students will identify the reciprocals of given sets of real numbers. Using examples such as 8/2 is the same as 8x1/2, students will discover the rule for division.

4 Outcome 3: (Linear Equations) Student will be able to: 1. Solve equations using properties of equality: a) One step equations involving a single operation b) Two step equations involving 2 operations c) Equations involving the need to combine like terms d) Equations involving grouping symbols and use of the distributive property e) Equations involving variables on both sides f) Literal equations and formulas g) Absolute value equations Students will be able to solve linear equations in a single variable over the real number system to include positive and negative integers, fractions, or decimals. 1. a-e Use balance scale as a model for an equation. The scale provides an illustration and reinforcement for the algebraic properties of equality. Begin with one step equations and extend models. 1. f. Use real life situations to reflect common formulas: such as, d=r t Manipulate the formula to solve for each variable; e.g., solve for rate given distance and time. 1. g. Use the number line to establish absolute value as a measure of distance regardless of direction. 1. a-e Students will work in pairs using algebra tiles to model an algebraic equation. Students will then write and solve equations modeled at various stations throughout the room. 1. f. Students will design a problem that will utilize a formula, create a solution and present the problem and the solution to the class. 1. g. Students will explain the sequence of solving absolute value equations as a function of distance. 2. Use applications: a) Solve word problems by developing equations using the 5 step plan b) Set up ad solve: i. Single number problems ii. Multiple number problems iii. Sum and difference problems iv. Perimeter and angle problems v. Consecutive integer problems vi. Motion problems vii. Coin mixture problems viii. Cost and income problems ix. Percent and interest problems x. Age problems 2. a. Create a word bank of terminology that will be needed. Create formula bank. Establish 5-step plan. 2. b. Use coin manipulatives to distinguish between number of coins and monetary value. Use diagrams to organize perimeter and angle problem information. Develop motion problem scenarios to describe same direction, opposite direction and round trip situations. 2. b. Use examples from the mall to illustrate percent applications. Students may bring advertisements from newspapers as well. Have students compare interest rates from local banks as an introduction to interest problems. 2. Given a definition of variables and an equation, students will be able to create a word problem scenario. Students will be able to solve a sampling of word problems using the 5-step plan.

5 Outcome: 4 (Inequalities) Students will be able to solve and graph linear inequalities in a single variable over the real number system. 1. Define: <,>, equal to or less then, equal to or greater than, not equal to 1. Demonstrate concept of >,< or = with algebra balance (with 4 pans 2 for positive weights and 2 for negative weights) 1. Have students demonstrate specific inequalities ( 8 > 3, etc.) by using a balance. 2. Graph inequalities on the number line. 2. Create two inequalities given any two points on the number line. Use temperature and racing speed as examples. 3. Properties of inequalities 3. Use Algebra balance. Relate to properties for equation similarities and differences. 2. Match inequalities with their corresponding graphs. 3. Show two equivalent inequalities. Ask students to identify what transformation was used to get from the first to the second (matching, multiple choice, etc.). 4. Solve and graph inequalities on the number line: a) Describe in set notation 5. Solve and graph compound inequalities: a) Intersection (conjunction) b) Union (disjunction) 4. Relate to solving and graphing linear equations similarities and differences. (Single element solution set vs. infinite element solution se.) e.g., (5) vs. (x :x >5). Compare graphs. 5. Explain the use of and and or using real life examples and Venn diagrams. Key words: Intersection = and Union = or Given the two inequalities x>1 and x<5; I=( x : x>1 and x<5) U = (x : x>1 or x<5) Demonstrate with graphs of different colors. Special case: a. empty set intersections φ (No graph) b. Whole number line unions all numbers 4. Solve an inequality, justifying each step. Then graph solution set on the number line. 5. Match solution sets with corresponding graphs.

6 Outcome: 5 (Graphing Linear Equations and Inequalities) The student will be able to 1. Define coordinate plane: a) Recognize and use appropriate terminology b) Plot points associated with given ordered pairs. Students will be able to graph equations and inequalities in the coordinate plane. 1. Students work in small groups to define one of the terms. Each group presents their definition and an example to the class. 1. Students will create a battleship field from 10 to 10 on a piece of graph paper. Students pair up and play. 2. Recognize and define relations and functions: a) Determine domain and range b) Use of function notation c) Establish criteria for a relation to be a function 2. Teacher draws a function machine explaining how input is the domain and output is the range. Use function notation to represent the changes in values from input to output. 2. Create a worksheet with pictures of graphs and points. Students will identify the domain, and the range and decide if the graph is a function or a relation. 3. Solve linear equation in 2 variables: a) Define standard form b) Recognize a linear equation as a relation c) Solve for domain or range d) Solve and graph using a table of values 3. Students work in pairs on a given linear equation (such as y= 3x+5), finding 4 ordered pairs that are solutions to the equation. Plot on a coordinate plane and graph. If available have students check their graphs using a graphing calculator. 3. Students create their own function rule for x and y. They complete a function table and graph the function on a coordinate plane. Students then trade function rules with a classmate and create a table and graph for this function. Assess by comparing results. 4. Determine the slope of a line: a) Define slope b) Link slope to rate of change c) Use slope formula 4. Teacher leads a discussion on a comparison of different ski slopes, rooflines or roller coasters. Introduce slope and rate of change. 4. Use 2 different colored dice. One is for x- values and one is for y values. Students roll dice. Plot the ordered pair. Students roll dice again and plot 2 nd ordered pair. Students then graph the line, and calculate the slope of the line.

7 Outcome 5: (continued) (Graphing linear equations and inequalities) Students will be able to graph equations and inequalities in the coordinate plane. 5. Relate slope to graph of a line a. Determine the slant/direction of the line given the slope b. Recognize the slope of vertical and horizontal lines c. Recognize parallel and perpendicular lines by their slope 5. Students are given 2 points and asked to: - graph the points - connect with a line - compare the slant of the line with the value of the slope. Extend this concept to horizontal, vertical, parallel and perpendicular lines. 5. At the blackboard, students sketch graphs of lines given various slope values. 6. Write the equation of a linear relation given: a) Point on the line and slope of the line b) Two points on the line c) Slope and y-intercept d) Point on the line and the equation of a parallel line *Point on the line and the equation of a perpendicular line 7. Graph a linear equation a) In slope-intercept form b) Using x and y-intercepts. 6 a-b Demonstrate the connection between the slope formula and the point-slope equation of a line. Demonstrate its use given a slope and a point or 2 points. c. Demonstrate the slope-intercept form of an equation and show how to graph. d-e Demonstrate how to find the slope given an equation. Use this information to write equations of parallel or perpendicular lines. 7. Students write the linear equation in slopeintercept form, y = mx+b. They identify the slope and the point that is the y-intercept, and then graph the equation using that point and the slope. 6. a-b. Students formulate an equation of a line given either a point and slope, or 2 points. c. Students graph lines on the blackboard given slopes and y intercepts. d-e. Give students pairs of equations. They must determine if lines are perpendicular, parallel or neither. 7. Students work in pairs helping each other to identify the slope and the y- intercept of several linear equations and then graph the lines accurately. 8. Graph a linear inequality 8. Teacher writes y = 2x-3 and y>2x-3 on the board. Each student is to find 1 point which satisfies 1 of the 2 sentences. Each student writes his/her point underneath the appropriate sentence. As a class, graph the equation and discuss how the inequality is different. 8. Students practice graphing and shading inequalities.

8 Outcome: 5 (continued) (Graphing Linear Equations and Inequalities) Students will be able to graph equations and inequalities in the coordinate plane. 9. Solve a system of 2 linear equations: a) By graphing b) By substitution c) By linear combination d) Use appropriate method to solve a word problem defined by a system of equations 9. Display 3 systems of equations on the board each demonstrating a different type of solution: i.e. parallel lines no solution, intersecting lines, 1 solution, and coinciding lines, infinite solutions. 9. Students graph 2 lines and discuss solution sets.

9 Outcome : 6 (Polynomials) Students will be able to perform addition, subtraction, multiplication and division of polynomial expressions. 1. Introduce polynomials: a). Define b). Classify by type and degree 1. Given examples of different polynomials, students will identify each according to the number of its terms, and then determine its degree. 1. Have students identify the type and degree of the polynomials on the board. 2. Simplify polynomial expressions 2. Have the students demonstrate how to recognize like terms in order to simplify polynomial expressions. 3. Add and subtract polynomials 3. For addition, have students identify like terms from both polynomials and then add. 2. At the board have students identify and underline the like terms, and then simplify the expression. 3. Simplify addition and subtraction of polynomials. 4. Multiply monomials: a). Product rule for exponent b). Power rule for exponents For subtraction, have students rewrite the second polynomial using the distributive property with 1, then identify and combine like terms. 4. Define Product and Power rule for exponents using examples such as (-3x²) (4x⁵) = (-3 4) (x²+⁵) and (-3x²)³ = (-3)³ (x²)³ = (-3)³ (x² ³) 4. Have students determine which statements are true by use of examples. 1). a ⁿ ⁿ=aⁿ aⁿ 2). aⁿ ⁿ =aⁿ +aⁿ 3). (ab)ⁿ ⁿ = aⁿ bⁿ 4). (a+b)ⁿ ⁿ =aⁿ bⁿ 5). aⁿ +ⁿ = aⁿ aⁿ 5. Multiply polynomial by a monomial 5. Demonstrate by use of the distributive property and product rule of exponents the multiplication of a polynomial by a monomial. 5. Send students to do sample problems at the board. 6. Multiply a binomial by a binomial: a) Vertical method b) FOIL method c) Binomial squared d) Sum and difference of 2 binomials 6. Students will work in pairs using tiles or manipulatives to demonstrate that (2x+5) (3x+2) has a rectangular area of 6x ²+ 19x Then demonstrate both vertical and FOIL methods for finding products of binomials. 6. Have students discover the rules for Squaring a binomial Sum and difference of two binomials

10 Outcome: 6 (continued) (Polynomials) Students will be able to perform addition, subtraction, multiplication and division of polynomial expressions. 7. Multiply trinomial by binomial 7. Use the distributive property or vertical method of multiplication to extend use to trinomials. 7. Have students practice simplifying trinomial multiplications. 8. Divide monomial by a monomial: a) Quotient rule for exponents b) Zero exponents c) Negative exponents d) Scientific notation 8. a. Define quotient rule for exponents 8. b.c. Using examples develop zero and negative exponent rules. 8. d. Demonstrate the use of scientific notation for writing large and small numbers. 8. Find examples of large and small numbers in newspapers and magazines to convert to scientific notation. 9. Divide polynomials by monomials 9. Use addition property of fractions with symmetric property of equality to show division of a polynomial by a monomial, i.e. 9. Have students rewrite polynomial division as a sum of quotients and simplify. a + b = a+b c c c therefore: a+b = a + b c c c

11 Outcome: 7 (Factoring) Students will be able to utilize a variety of factoring techniques to solve polynomial equations. 1. Find prime factorization of integers: a) Define prime and composite b) Identify primes Use factor trees to determine prime factorization of integers. 1. Students will be able to determine primes from by recreating the Sieve of Eratosthenes. 2. Find greatest common factor: a) Given 2 or more integers b) Given 2 or more monomials 3. Factor a) Greatest monomial factor of a polynomial b) Difference of 2 squares c) Trinomials d) Grouping with 4 or more terms e) Combined factoring 4. Solve equations by factoring: a) Quadratic equations b) Higher degree equations 5. Solve word problems a) Area b) D=RT 2.a. Use prime factorization to establish GCF of 2 or more integers. 2. b. Use algebra tiles to determine monomial GCF. 3. Use algebra tiles to form a rectangle. The dimensions of the rectangle will be the factors of the polynomial. Create and use a flow chart to describe factoring sequence. 4. Use the graphing calculator to illustrate that the roots of a polynomial equation are the x intercepts of the graph. 5. Model real life situations utilizing quadratic equations. Ask students to determine the placement of an area rug in a room measuring 9m x 12m so that a uniform strip will remain uncovered. Have students determine the height of an arrow shot into the air given its rate and time, using h=vt-16t² 2. Student will be able to determine GCF of higher order expressions such as (132x⁴, 220x⁵, and 88x⁷) 3. Students will be able to match trinomial to one (or both) of its factors in a matching game. Students will be able to link FOIL and factoring together in a round-robin game. 1) Each student selects 2 factors. 2) Papers are passed. Students FOIL. 3) Papers are passed. Students factor. 4) Process repeated until each student has completed all problems. 5) Compare original problems to outcomes. 4. Students will work in pairs. Each student will solve 5 quadratic equations. After solving, students will exchange paper and check by use of graphing calculators. *Students will play a card game by matching quadratic equations to their roots. 5. Students will work in groups of 4. Each will be responsible for problems of varying types. Students must establish a sketch, identify and solve the equation and express answer in terms of the problems. Each group will be called upon to present one problem.

12 Outcome: 8 (Rational expressions) Students will multiply, divide and simplify rational expressions, solve rational equations, and use proportions to solve word problems. 1. Identify and define a rational expression. 1. Provide examples of rational expressions using simple numerical fractions and fractions containing polynomials. 1. Rewrite division problems as rational expressions. (For example 2 5 as 2/5.) 2. Determine the values for which the expression is undefined. 3. Simplify rational expressions using factoring and cancellation properties. 2. Introduce the general rule: If a/b = c, then b c= a (b 0). Show why 6/0 must be undefined (or meaningless). Introduce rational expressions containing polynomial denominators and discuss when these expressions would be undefined. (Set denominator = to zero and solve). Connect this concept to domain of a function. 3. Use a 2 step procedure: - Factor numerator and denominator completely. - Divide out common factors in numerator and denominator. Give examples of when not to divide. (e.g., in 2/2x-1 2 s cannot be divided out) 2. Find the domain of given rational expressions. 3. Give several examples of correctly and incorrectly simplified rational expressions. Students identify the correct ones and amend the incorrect ones. 4. Multiply rational expressions. 4. Introduce the general rule: (a/b)(c/d)= ac/bd (b 0, d 0). When multiplying rational expressions containing polynomials, use 3 step procedure: a. Factor numerators and denominators completely. b. Divide out common factors in numerators and denominators. c. Multiply numerators. Multiply denominators. 4. Multiply rational expressions containing an assortment of monomial and polynomial numerators and denominators. Answer should be in simplified form. Students should be able to explain their steps.

13 Outcome: 8 (continued) (Rational expressions) 5. Divide rational expressions using reciprocals. 5. Introduce the general rule: (a/b) (c/d)=(a/b)(d/c) (b 0), c 0, d 0). (Division equals multiplication by reciprocal.) When dividing rational expressions containing polynomials, use 4- step procedure: Students will simplify rational expressions, rational equations and use proportions to solve word problems. - Rewrite division problem as multiplication (use reciprocal) Repeat steps a,b,c in #4. 5. Divide rational expressions and be able to explain steps. Give examples that were incorrectly done. Have students identify what went wrong. 6. Divide polynomials using long division. 6. Show examples of division of a polynomial by a polynomial. Review the long division algorithm for integers. Relate the algorithm to division of a polynomial by a binomial. Give examples. (Use area concept from geometry.) 6. Divide polynomials by monomials. Describe steps for the long division algorithm. Use long division to divide a polynomial by a binomial. 7. Find Least Common Denominator for rational expressions. 8. Add or subtract rational expressions using Least Common Denominator. Optional strategies: Model long division by use of algebra tiles and/or synthetic division. 7. Review procedure for finding LCM of monomial expressions. Find the LCM of two (or more) polynomial expressions that: - a) Share common factors - b) Have no common factors LCD = LCM of the denominators. 8. Introduce the General Rule: a/b+c/b= a+c (b 0). b Give addition and subtraction examples using monomial denominators and more complex polynomial denominators. Use 3-step procedure: - a) Create equivalent rational expressions using the LCD - b) Add (or subtract numerators of these equivalent rational expressions. c) (Denominator is LCD) - Simplify, if possible 7. Student should be able to find the LCD of rational expressions containing an assortment of polynomial denominators and explain the difference between a common denominator and the least common denominator. 8. Students should be able to find the perimeter of a polygon given the lengths of the sides in rational expression form.

14 Outcome: 8 (Rational expressions) Students will simplify rational expressions, rational equations and use proportions to solve word problems. 9. Identify ratios and solve proportions: a) Solve word problems involving ratio and proportion 10. Solve rational equations: a) Fractional coefficients b) Fractional equations c) Determine extraneous solutions 9. Consider ratios as rational expressions, and proportions as equations involving two rational expressions. Introduce Cross Product Property: If a/b = c/d, then ad=bc (b 0, d 0). Demonstrate the usefulness of this property in solving word problems in which a proportion is involved. (Scale drawings, population problems, etc.) 10.a-b. Multiply both sides of the equation by the LCD and solve the resulting equation by standard techniques. c. Test solutions in original equation to validate. (f any solution yields a denominator of zero, it is extraneous and must be discarded) 9. Students should be able to create and solve word problems involving proportions. 10. Given a specific replacement set, students should be able to identify an extraneous solution.

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