Note: Problem Solving Algebra Prep is an elective credit. It is not a math credit at the high school as its intent is to help students prepare for Algebra by providing students with the opportunity to develop the skills necessary to be successful in a high school algebra course. For this reason, it addresses many of the 7 th and 8 th grade standards and benchmarks. Subject Area Mathematics Senior High Course Name Problem Solving Algebra Prep Date June 2009 Unit Content Standards Addressed Skills/Benchmarks Essential Questions Assessments 12 Arithmetic with Letters Arithmetic and Algebra Representing Numbers Using Letters Integers on the Number Line Adding Integers Subtracting Integers Multiplying Integers Dividing Positive and Negative Integers Simplifying Expressions one variable Simplifying Expressions several variables Positive Exponents Formulas with Variables Grade 7 Number & Operation 7.1.1 Read, write, represent and compare positive and negative rational numbers, expressed as integers, fractions and decimals. 7.1.2 Calculate with positive and negative rational numbers, and rational numbers with whole number exponents, to solve real-world and mathematical problems. 7.2.4 Represent real-world and mathematical situations using equations with variables. Solve equations symbolically, using the properties of equality. Also solve equations graphically and numerically. Interpret solutions in the original context. 7.1.1.3: Locate positive and negative rational numbers on the number line. Understand the concept of opposites. 7.1.2.1: Add, subtract, multiply, and divide positive and negative integers. 7.1.2.5: Use the symbol for absolute value. Demonstrate of absolute value with respect to distance. 7.2.4.1: Represent relationships in various contexts with equations involving variables and positive and negative rational numbers. Use the properties of equality to solve for the value of a variable. Interpret the solution in the original context. For example: Solve for w in the equation P = 2w + 2l when P = 3.5 and l = 0.4. What is the difference between numerical and algebraic expressions? How can unknown quantities be expressed? How are arithmetic operations performed with positive and negative numbers? When and how can variable expressions be simplified? How are variable expressions used to solve real world problems? Another example: To post an Internet website, Mary must pay $300 for initial set up and a monthly
8 th Grade Algebra 8.2.4 Represent real-world and mathematical situations using equations and inequalities involving linear expressions. Solve equations and inequalities symbolically and graphically. Interpret solutions in the original context. fee of $12. She has $842 in savings, how long can she sustain her website? 8.2.4.2: Solve multi-step equations in one variable. Solve for one variable in a multi-variable equation in terms of the other variables. Justify the steps by identifying the properties of equalities used. 1. Recognize numerical and algebraic expressions. 2. Understand the use of variables in algebraic expressions. 3. Understand positive and negative integers, opposites, and absolute value. 4. Discover and use rules related to adding, subtracting, multiplying, and dividing integers. 5. Simplify expressions with one or more variables. 6. Read and write exponents. 7. Use formulas with variables. 12 The Rules of Arithmetic Commutative Property of Addition Commutative Property of Multiplication Associative Grade 7 Algebra 7.2.3 Apply of order of operations and algebraic properties to generate equivalent numerical and algebraic expressions containing positive and negative rational numbers and grouping symbols; evaluate such expressions. 7.2.3.1: Use properties of algebra to generate equivalent numerical and algebraic expressions containing rational numbers, grouping symbols and whole number exponents. Properties of algebra include associative, commutative and distributive laws. Why did the calculator give me the wrong answer? Why does the order in which operations are performed often change the outcome? How does the use of mathematical properties simplify or make some
Property of Addition Associative Property of Multiplication The Distributive Property Multiplication The Distributive Property Factoring Properties of Zero Properties of 1 Powers and Roots Order of Operations 7.2.3.3: Apply of order of operations and grouping symbols when using calculators and other technologies. 1. Use the Commutative Property of Addition & Multiplication. 2. Use the Associative Property for Addition & Multiplication. 3. Use the Distributive Property over multiplication and to factor. 4. Apply the Property of Zero. 5. Apply the Property of 1 6. Raise expressions to integer powers. 7. Find square root of perfect squares. 8. Apply Order of Operations to simplify and/or evaluate expressions. problems easier to solve? 18 Linear Equations with One Variable Writing Equations Solving Equations: x-b=c Solving Equations: x+b=c Solving Multiplication Equations Solving Equations with Fractions Solving Equations with More than 8 th Grade Algebra 8.2.4 Represent real-world and mathematical situations using equations and inequalities involving linear expressions. Solve equations and inequalities symbolically and graphically. Interpret solutions in the original context. 8.2.4.3: Express linear equations in slope-intercept, point-slope and standard forms, and convert between these forms. Given sufficient information, find an equation of a line. For example: Determine an equation of the line through the points (-1,6) and (2/3, -3/4). 8.2.4.4: Use linear inequalities to represent relationships in various contexts. Without guessing, how can the solution to an equation be found using concepts of algebra? How can the sides of a right triangle be found? How is and interval expressed using mathematical symbols? What real world applications ask for an interval solution?
One Step Equations Without Numbers Formulas The Pythagorean Theorem Inequalities on the Number Line Solving Inequalities in One Variable Using Equations For example: A gas station charges $0.10 less per gallon of gasoline if a customer also gets a car wash. Without the car wash, gas costs $2.79 per gallon. The car wash is $8.95. What are the possible amounts (in gallons) of gasoline that you can buy if you also get a car wash and can spend at most $35? 8.2.4.5: Solve linear inequalities using properties of inequalities. Graph the solutions on a number line. For example: The inequality -3x < 6 is equivalent to x > -2, which can be represented on the number line by shading in the interval to the right of -2. 1. Solve one-step equations involving addition, subtraction, or multiplication. 2. Solve multi-step equations involving the four basic operations. 3. Solve for a variable. 4. Use the Pythagorean Theorem to solve for an unknown side. 5. Represent inequalities on a Number Line. 6. Solve inequalities in one variable. 14 Applications of Algebra 7 th Grade Algebra Standard 7.2.2 Recognize proportional 7.2.2.1: Represent proportional How can percentages be estimated without the use of a
Writing Equations Odd and Even Integers Using the 1% Solution to Solve Problems Using the Percent Equation Solving Distance, Rate, & Time Problems Using a Common Unit--Cents Calculating Simple Interest Deriving a Formula for Mixture Problems Ratio and Proportion Using Proportions relationships in real-world and mathematical situations; represent these and other relationships with tables, verbal descriptions, symbols and graphs; solve problems involving proportional relationships and explain results in the original context. relationships with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another. Determine the unit rate (constant of proportionality or slope) given any of these representations. For example: Larry drives 114 miles and uses 5 gallons of gasoline. Sue drives 300 miles and uses 11.5 gallons of gasoline. Use equations and graphs to compare fuel efficiency and to determine the costs of various trips. 7.2.2.2: Solve multi-step problems involving proportional relationships in numerous contexts. For example: Distance-time, percent increase or decrease, discounts, tips, unit pricing, lengths in similar geometric figures, and unit conversion when a conversion factor is given, including conversion between different measurement systems. calculator? How do percents relate to a portion of the circle? A trip takes more time to return than it did to get to the destination. How is the average rate for the entire trip calculated? What is meant by simple interest, and how is it calculated? Three different dried fruits are mixed to make a snack pack. How should the price to sell the snack pack be determined? Another example: How many kilometers are there in 26.2 miles? 7.2.2.3: Use knowledge of proportions to assess the reasonableness of solutions. For example: Recognize that it would be unreasonable for a cashier to request $200 if you purchase a $225 item at 25% off.
7.2.4 Represent real-world and mathematical situations using equations with variables. Solve equations symbolically, using the properties of equality. Also solve equations graphically and numerically. Interpret solutions in the original context. 7 th Grade Data Analysis and Probability 7.4.2 Display and interpret data in a variety of ways, including circle graphs and histograms. 7.2.4.2: Solve equations resulting from proportional relationships in various contexts. For example: Given the side lengths of one triangle and one side length of a second triangle that is similar to the first, find the remaining side lengths of the second triangle. Another example: Determine the price of 12 yards of ribbon if 5 yards of ribbon cost $1.85. 7.4.2.1: Use reasoning with proportions to display and interpret data in circle graphs (pie charts) and histograms. Choose the appropriate data display and know how to create the display using a spreadsheet or other graphing technology. 1. write an algebraic equation for a number sentence. 2. identify formulas to use in specific types of problems. 3. write problems using algebraic formulas. 4. solve problems by applying algebraic equations. 10 Data, Statistics, & Probability Organizing Data Range, Mean, Median, & Mode Box-and-Whiskers 7 th Grade Data Analysis & Probability 7.4.1 Use mean, median and range to draw conclusions about data and make predictions. 7.4.1.1: Design simple experiments and collect data. Determine mean, median and range for quantitative data and from data represented in a display. Use these quantities to draw How is data used to make decisions? What does the center of the data mean? How are probabilities
Plots The Probability Fraction Probability and Complementary Events Tree Diagrams and Sample Spaces Dependent and Independent Events The Fundamental Principle of Counting Multistage Experiments 7.4.3 Calculate probabilities and reason about probabilities using proportions to solve real-world and mathematical problems. conclusions about the data, compare different data sets, and make predictions. For example: By looking at data from the past, Sandy calculated that the mean gas mileage for her car was 28 miles per gallon. She expects to travel 400 miles during the next week. Predict the approximate number of gallons that she will use. 7.4.3.2: Calculate probability as a fraction of sample space or as a fraction of area. Express probabilities as percents, decimals and fractions. calculated? What is the difference between the probability of blue and green and the probability of blue or green? How is probability used to make decisions? For example: Determine probabilities for different outcomes in game spinners by finding fractions of the area of the spinner. 7.4.3.3: Use proportional reasoning to draw conclusions about and predict relative frequencies of outcomes based on probabilities. For example: When rolling a number cube 600 times, one would predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. 1. organize data into graphs. 2. read and interpret graphic representations
3. determine range and measures of central tendency. 4. compute probabilities and complementary event involving statistics. 9 Linear Equations and Inequalities in the Coordinate plane The Coordinate System Graphing Equations Intercepts of Lines Slopes of Lines Writing Linear Equations Lines as Functions Domain and Range of a Function 7 th Grade Algebra Standard 7.2.2 Recognize proportional relationships in real-world and mathematical situations; represent these and other relationships with tables, verbal descriptions, symbols and graphs; solve problems involving proportional relationships and explain results in the original context. 8 th Grade Algebra: 8.2.1 Understand the concept of function in real-world and mathematical situations, and distinguish between linear and nonlinear functions. 7.2.2.4: Represent real-world or mathematical situations using equations and inequalities involving variables and positive and negative rational numbers. For example: "Four-fifths is three greater than the opposite of a number" can be represented as 4 = n + 3, and "height no bigger 5 than half the radius" can be represented as h r. 2 Another example: "x is at least -3 and less than 5" can be represented as 3 x < 5, and also on a number line. 8.2.1.1: Understand that a function is a relationship between an independent variable and a dependent variable in which the value of the independent variable determines the value of the dependent variable. Use functional notation, such as f(x), to represent such relationships. What is meant by slope? What is a function? What does domain and range have to do with independent and dependent variables? What does a shaded region in the coordinate plane mean? What might the graph of distance traveled with respect to time over a week-long trip look like? For example: The relationship between the area of a square and the side length can be expressed as
f() x = x2. In this case, f (5) = 25, which represents the fact that a square of side length 5 units has area 25 units squared. 8.2.1.2: Use linear functions to represent relationships in which changing the input variable by some amount leads to a change in the output variable that is a constant times that amount. For example: Uncle Jim gave Emily $50 on the day she was born and $25 on each birthday after that. The function f( x) = 50 + 25xrepresents the amount of money Jim has given after x years. The rate of change is $25 per year. 8.2.1.3: Understand that a function is linear if it can be expressed in the form f () x = mx + b or if its graph is a straight line. For example: The function f() x = x2 is not a linear function because its graph contains the points (1,1), (-1,1) and (0,0), which are not on a straight line. 8.2.2 Recognize linear functions in real-world and mathematical situations; represent linear functions and other functions with tables, verbal descriptions, symbols and graphs; solve problems involving these functions and 8.2.2.1: Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another. 8.2.2.2: Identify graphical properties of linear functions including slopes
explain results in the original context. and intercepts. Know that the slope equals the rate of change, and that the y-intercept is zero when the function represents a proportional relationship. 8.2.2.3: Identify how coefficient changes in the equation f (x) = mx + b affect the graphs of linear functions. Know how to use graphing technology to examine these effects. 1. identify the parts of a graph. 2. locate and plot points in the coordinate system. 3. solve equati0ns for ordered pairs and graph a line. 4. find the x-intercept and y- intercept of a graph. 5. determine the slope of a line 6. write and solve an equation of a straight line. 7. identify and evaluate functions. 8. determine the range of a function with a given domain. 9. graph inequalities. 10. interpret and create graphs without numbers. 10 Geometry Angles and Angle Measure Pairs of Lines in Planes and in Space 7 th Grade Algebra Standard 7.2.1 Understand the concept of proportionality in realworld and mathematical situations, and distinguish between proportional and other relationships. 7.2.1.2: Understand that the graph of a proportional relationship is a line through the origin whose slope is the unit rate (constant of proportionality). Know how to use graphing technology to examine How is geometry used in carpentry or architecture? How are angles labeled to avoid confusion where more than one angle is pictured?
Angles Measures in a Triangle Naming Triangles Quadrilaterals Congruent and Similar Triangles 7.3.2 Analyze the effect of change of scale, translations and reflections on the attributes of two-dimensional figures. 8 th Grade Geometry and Measurement 8.3.1 Solve problems involving right triangles using the Pythagorean Theorem and its converse. what happens to a line when the unit rate is changed. 7.3.2.1: Describe the properties of similarity, compare geometric figures for similarity, and determine scale factors. For example: Corresponding angles in similar geometric figures have the same measure. 8.3.1.1: Use the Pythagorean Theorem to solve problems involving right triangles. For example: Determine the perimeter of a right triangle, given the lengths of two of its sides. How can triangles be classified? What is needed for two triangles to be congruent? What is meant by similar geometric figures? How can a quadrilateral be divided into two triangles? Another example: Show that a triangle with side lengths 4, 5 and 6 is not a right triangle. 1. name and determine the measure of angles. 2. identify how lines are related in planes and space. 3. use theorems to help solve problems involving triangles. 4. name triangles by their characteristics. 5. determine the measures of angles in quadrilaterals. 6. use theorems to determine whether triangles are congruent or similar.