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EQUATIONS and INEQUALITIES Linear Equations and Slope Overview of Objectives, students should be able to: 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line perpendicular to a given line. 2. Slope-intercept a. Convert a linear equation into the slopeintercept form, y = mx + b b. Identify the slope and y-intercept of an equation using slope-intercept form. 3. Point-slope formula a. Use the point slope formula to find the equation of a line given the slope and a point. b. Use the point slope formula to find the equation of a line given two points. c. Find equations of parallel and perpendicular lines using the point slope formula given varied information. 4. Standard a. Convert a linear equation into standard form: Ax + By = C b. Graph equations of the form x = a, where a is a constant c. Graph equations of the form y = b, where b is a constant Main Overarching Questions: 1. How do you find the slope of a line? 2. What is the relationship between slopes of parallel lines? …of perpendicular lines? 3. How do you find the slope and y-intercept of a line given its equation? 4. How do you write the equation of a line given slope and a point on the line? …given 2 points on the line? 5. How do you graph equations in the form x = a and y = b, where a and b are constants? The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. 5. Use the point-slope form to solve problems. Objectives: • Slope o Calculate the slope of a line given two points o Calculate the slope of a line parallel to a given line. o Calculate the slope of a line perpendicular to a given line Activities and Questions to ask students: 1. Where have you seen the concept of slope used in the real world? What does the word mean to you? 2. Demonstrate graphically the meaning of slope, rise/run: rise as the vertical distance between two points, run as the horizontal distance. 3. How can we mathematically compute these differences? 4. How can we write a formula to compute the slope for any two points, (x1, y1) and (x2, y2)? 5. Have students plot points to create two parallel lines on a coordinate plane. Label the points and use the slope formula to compute the slope between two points on both lines. 6. What do the two lines have in common? 7. What can you conclude about the slopes of parallel lines? Present graphs of perpendicular lines with at least 2 points marked on each line.* 8. Have students compute slope of each line and ask if they can draw a conclusion about the slopes of perpendicular lines. *It may be necessary to repeat this exercise with other pairs of perpendicular lines until students can discover the relationship of the slopes( negative reciprocals). • • Slope-intercept o Convert a linear equation into the slopeintercept form, y = mx + b o Identify the slope and y-intercept of an equation using slope-intercept form. Point-slope formula o Use the point slope formula to find the equation of a line given the slope and a point. o Use the point slope formula to find the equation 1. Why is the form y = mx + b called slope-intercept form? Which letter represents the slope and which represents the y-intercept? Where have you seen this letter used previously? 2. Slope-intercept form is solved for which variable? 3. How do you solve an equation for y? Show how the point-slope form is derived from the slope formula. 1. Why do you think this formula is called the “point-slope” form? 2. If we are given the slope of a line and a point on the line, how can we use this formula to write an equation of the line? The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. o of a line given two points. Find equations of parallel and perpendicular lines using the point slope formula given varied information. 3. If we are given two points on a line, what must we find before we can use the point- slope formula? How do we find the slope given two points? 4. Consider what we learned about the slopes of parallel and perpendicular lines. (Review if necessary.) How can we write the equation of a line through a given point IF we are given an equation of a line parallel to it? What information is needed from the given equation? 5. How can we write the equation of a line though a given point IF we are given an equation of a line perpendicular to it? What information is needed from the given equation? • Standard o Convert a linear equation into standard form: Ax + By = C o Graph equations of the form x = a, where a is a constant o Graph equations of the form y = b, where b is a constant 1. Review previous forms of linear equations that have been discussed. Ask students to compare the forms. How is standard form different? 2. Can you determine the slope and y-intercept when an equation is in standard form? Clarify the requirements for A,B,C (some textbooks do not require A to be positive, etc.) 3. What steps do we take to change y = 2x + 5 to standard form? 4. What steps do we take to change y = ½ x – 3 to standard form? Group work or teacher led discussion: 5. Have students list a few ordered pairs with x coordinate = 3, such as (3, -1), (3, 0), (3, 2), (3, 4), and graph the points on a coordinate plane. What type of line is formed when these points are connected? What do all these points have in common? (x-co-ordinate = 3) What can we call this line if all x-coordinates are 3? 5. Show students graphs of other vertical lines without plotting points and ask for the equation of each line. 6. Have students repeat the exercise above with a common y-coordinate and repeat questions. 7. Have students summarize how to graph horizontal and vertical lines. The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. • Use the point-slope form to solve problems 1. When solving problems, what information is needed to use the point-slope form? Systems of Equations Overview of Objectives, students should be able to: 1. Determine if a given ordered pair is a solution to a system of linear equations. 2. Solve systems of linear equations using graphing. 3. Solve systems of linear equations by substitution. 4. Solve systems of linear equations by addition. 5. Select the most efficient method for solving a system of linear equations. 6. Identify systems that have no solution or infinitely many solutions. Objectives: • Determine if a given ordered pair is a solution to a system of linear equations. Main Overarching Questions: 1. When is an ordered pair a solution of a system? 2. How do you solve systems of equations by graphing? 3. How do you solve systems of equations by algebraic methods? 4. Compare and contrast the methods of solving systems for efficiency and accuracy. 5. How do you determine when a system has no solution or infinitely many solutions? Activities and Questions to ask students: 1. Present two linear equations and have students substitute an ordered pair into x and y. What is meant by the term “solution to an equation”? Does the ordered pair create true or false statements? Is this point a solution for both of these equations? 2. When is an ordered a solution of a system of linear equations? • Solve systems of linear equations using graphing. Group activity or teacher directed: 1. Students will graph 3 systems of linear equations: a) a pair of intersecting lines b) a pair of parallel lines c) two equations that are the same line ( coinciding lines according to some texts) 2. Direct students to compare the three systems: a) describe the type of lines b)describe how The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. many points they have in common c) compare the equations within each system 3. Groups may present their results. 4. Summarize using 3-column note format. • Solve systems of linear equations by substitution. 1. What does substitution mean? What can be substituted without changing the solution of an equation? 2. If 2 equations are solved for y like y = 2x – 1 and y = x + 4, can you say that 2x-1 = x + 4. Why or why not? Now that you can solve for x, how can you find y? For students who struggle with substitution method, try the above method. 3. Demonstrate technique of solving one equation for a variable. How can we use substitution to combine these two equations into a single equation with one variable? 4. How does this help us solve the system? How do you find the second variable? 5. What happens if you substitute into the wrong variable? • Solve systems of linear equations by addition • Select the most efficient method for solving a system of linear equations. • Identify systems that have no solution or infinitely many solutions. 1. How do you add two equations? What parts can you add? 2. Ask students to add two given equations (where a variable will cancel). How does this help us solve the system? How do you find the second variable? 3. Ask students to add two given equations where a variable does not cancel? Does this help us solve the system? Why not? What needs to happen? What is it about the coefficients that makes a variable cancel when adding? 4. How can we change an equation so that a variable will cancel when we add? IF we change one part of the equation, what must be done to the remaining terms in the equation? 5. Introduce or reemphasize term: equivalent equations 1. Have students compare/contrast the 3 methods of solving a system. Do lines always intersect at integer points? Can you always read the coordinates of the intersection on a graph? Can you determine fraction solutions when solving using algebraic methods. 2. Ask students to make a conclusion about the efficiency and accuracy of each method. 3. Give students a system with no solution and ask them to use either the substitution or addition method to solve. What happens to the variables? What kind of statement is left? Is it true or false? Ask students to solve each equation for y and compare the slopes and yintercepts. What kind of lines are they and how many solutions are there for this system? 4. Give students a system with infinitely many solutions and ask them to solve using addition or The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. substitution method. What happens to the variables and the constants? What statement is left? True or false? Ask students to solve both equations for y and to make a conclusion about the type of lines and the number of solutions. Problem Solving Overview of Objectives, students should be able to: 1. Setup and solve a system of linear equations to solve dual investments with simple interest. 2. Setup and solve a system of linear equations to solve mixture problems. 3. Find the profit function and cost function. 4. Find the break-even point. Objectives: • Setup and solve a system of linear equations to solve dual investments with simple interest. • Setup and solve a system of linear equations to solve mixture problems. Main Overarching Questions: 1. How can systems of equations be used to solve real world equations? 2. What does “break-even” mean (in regard to profit and cost) and how do you determine this break-even point on a graph? Activities and Questions to ask students: 1. What information is given about the amount of money invested? Ask students to write an equation using x and y for the amounts invested in the two accounts. 2. What information is given about the total interest earned? How is simple interest calculated? What expressions would be used to calculate interest for x and y amounts? 3. Ask students to write an equation using x and y and the given interest rates 4. showing how the total interest was earned. 5. What method would you use to solve this system? 6. 5. Ask students to solve and check the system. 1. What information is given about the total amount of mixture needed? Ask students to write an equation using x and y for the amount of each ingredient to be mixed. 2. What information is given about the percent of each ingredient and the percent of the total mixture? Ask students to write an equation using this information. 3. What method would you use to solve this system? 4. Ask students to solve and check the system. The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. • Find the profit function and cost function. • Find the break-even point. 1. What is profit? How is it calculated? 2. What is cost and how is total cost calculated? 3. Provide students with numbers of sales revenue and costs for an item and ask them to compute profit for each example. Ask students to substitute words for the numbers and write a formula for computing profit and cost. 1. Ask students graph profit and cost functions and find the point of intersection. Ask students to explain what the break-even point means in terms of profit and cost. Absolute Value Overview of Objectives, students should be able to: 1. Solve absolute value equations of the form |Ax + B | =C 2. Solve absolute value equations of the form |Ax + B | +C=D 3. Solve absolute value equations of the form |Ax + B | = |Cx + D| Objectives: Main Overarching Questions: 1. What does absolute value mean? 2. How do you solve absolute value equations? Activities and Questions to ask students: • Solve absolute value equations of the form |Ax + B | =C 1. What is absolute value? Can more than one number have the same absolute value? Give an example and explain your reasoning. What number is the exception? 2. For |x| = 5, what numbers could replace the x? How many solutions are there? Write equations for these solutions. 3. For |x + 1| = 5, what two numbers could x be? Using our equations for the example above, write two equations to solve this problem. 4. Write an absolute value equation where you might have only one solution. 5. Write an absolute value equation where you might have no solution. • Solve absolute value equations of the form |Ax + B | 1. Is C a part of the absolute value in this equation? How can we get rid of C and simplify this problem so that our absolute value is isolated? The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. • +C=D 2. What needs to be done first to solve |2x + 3|+ 4 = 9? Solve and check. Solve absolute value equations of the form |Ax + B | = |Cx + D| 1. How is this equation different from previous absolute value equations? 2. If |x| =|2| what must be true about x? Must x = 2? Write 2 equations for x. 3. If x = 2 or x = -2, then how can we solve an equation like |x+2|=|2x – 3| using opposites? 4. Ask students to solve and check both answers. General Knowledge of Inequalities Overview of Objectives, students should be able to: 1. Use interval notation to represent solutions to inequalities in one variable. 2. Use graphing to represent solutions to inequalities in one variable. 3. Solve linear inequalities in one variable. 4. Recognize linear inequalities that have no solution or infinitely many solutions. 5. Provide answers is set notation. Objectives: • Use interval notation to represent solutions to inequalities in one variable. Main Overarching Questions: 1. How do you solve and graph inequalities in one variable? 2. How do you determine the number of solutions to a linear inequality or if no solution exists? 3. How do you use set notation and interval notation to express the solutions of linear inequalities? Activities and Questions to ask students: 1. If x > 3, what number(s) does this x stand for? What is the least number included? Is 3 included? Why or why not? 2. We use (3, oo) to show all real numbers great than 3. How can we show all real numbers > 7? … > -2? 3. We use (-oo, 4) to show all real numbers less than 4. How do you use interval notation to show all reals < 8? …< -3? 4. If x is greater than OR EQUAL TO 5, then using interval notation we write [5, oo) and if x is less than OR EQUAL TO 2, we write (-oo,2]. How do you express all reals less than or equal to 7 in interval notation? The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. • Use graphing to represent solutions to inequalities in one variable. • Solve linear inequalities in one variable. • Recognize linear inequalities that have no solution or infinitely many solutions. • Provide answers in set notation 1. Present graph of x > 5. How is x > 5 represented on this number line? Is 5 included in the graph? How is this shown? 2. Present graph of x > or = 5. Is 5 included in this graph? How is this shown? 3. Repeat with > and < or = to if needed. 1. Review properties used in solving equations. 2. How would we solve x + 2 = 6? How can we use these properties to solve X + 2 < 6? 3. How is the solution to an inequality different from a solution to an equation? How can we check our solutions? 4. Ask students to solve -3x < 9 and check their solution. If students fail to change < to >, ask why their solution does not work when checked? 5. Or ask students to divide or multiply both sides of a true inequality like 2< 4 by -1. Is this still a true inequality? Why or why not? 6. What must be done to make the solution of -3x < 9 work? After what other operation will you need to switch the inequality sign? 1. Can you think of a number for x so that x > x + 5 is a true statement? Try a positive, a negative, and zero. Will x + 5 always be greater than x? How many solutions will this inequality have? 2. Can you think of a number for x so that x < x + 5 is a true statement? Can you think of more numbers? How many solutions will this inequality have? 1. {x|x> 6} is an example of set notation. It translates “the set of all x, such that x is greater than 6”. How would you express all real numbers less than or equal to -5 using set notation? How would you express all real numbers greater than 4 using set notation? Solve Compound Inequalities Overview of Objectives, students should be able to: 1. Solve compound inequalities involving “and”. 2. Solve compound inequalities involving “or”. Main Overarching Questions: 1. How do you solve compound inequalities? The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. 3. Solve compound inequalities of the form Ax + B < Cx + D < Ex + F Objectives: • Solve compound inequalities involving “and”. Activities and Questions to ask students: 1. What is a compound sentence? 2. What numbers do x > 4 and x < 8 have in common? What numbers are between 4 and 8? Are the answers the same for both questions? 3. Graph the answer to these questions. • Solve compound inequalities involving “or”. 1. If the solutions to an equation are x = 3 or x = 4, what does this mean? What “and” and “or” between two math statements mean? does 2. Ask students to solve x + 2 > 6 or x + 2 < 2 and graph both on a number line. Do numbers from either solution make the statement true? Why or why not? 3. How are “and” and “or” statements different? How are their solutions different? Compare their graphs. 2. Solve compound inequalities of the form Ax + B < Cx + D < Ex + F 1. Recall that “and” statements and “between statements” are the same. If 5<x<8, what numbers could x represent? Can these numbers be described as x > 5 and x < 8? 2. How can you rewrite 5 < x+1 < 8 as two statements? Ask students to rewrite as an “and” statement and solve both inequalities. 3. How can you rewrite x + 2 < 3x - 1 < 2x + 6 as two inequalities? Ask students to rewrite and solve. The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. Solve Absolute Value Inequalities Overview of Objectives, students should be able to: 1. Solve absolute value inequalities of the form |Ax + B| < C, where C > 0. 2. Solve absolute value inequalities of the form |Ax + B| > C, where C > 0. 3. Solve absolute value inequalities of the form n|Ax + B| > C, where C > 0 or C < 0 and n is any real number. 4. Recognize absolute value inequalities with no solution or all real numbers as solutions. Objectives: • • 1. How do you solve absolute value inequalities? 2. How do you determine if an absolute value inequality has no solution or infinitely many solutions? Activities and Questions to ask students: Solve absolute value inequalities of the form |Ax + B| < C, where C > 0. 1. Main Overarching Questions: Solve absolute value inequalities of the form |Ax + B| > C, where C > 0. Solve absolute value inequalities of the form n|Ax + B| > C, where C > 0 or C < 0 and n is any real 1. Direct students to draw a number line and label and graph numbers whose absolute value is less than 5. If students graph only integers or only positive numbers, ask them if -2.5 has an absolute value < 5. 2. Ask students to draw a conclusion about the numbers whose absolute value is less than 5. 3. Ask students to write an “and” or “between” statement to describe their conclusion for all numbers, x, such that |x| < 5. 4. Using this interpretation, how can we solve |x +2|< 5? 1. Direct students to draw a number line and label and graph numbers whose absolute value is greater than 3. |x| > 3 2. Ask students to describe the solutions using inequalities and then write as a compound inequality. Discuss why the word “or” is used instead of “and”. 3. How is the solution to a “>” problem different from a “<” problem? 4. Can you use a “between” statement for this problem? Why or why not? 5. Using a compound inequality, how can we solve |x + 2| > 3? 1. For C > 0, n > 0: Solve 3|x-1|> 6 and check solutions by substitution. 2. For C > 0, n < 0: Solve -3|x-1| > 6 . What happens when we divide both sides by -3? The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. number. 3. 4. 5. • Recognize absolute value inequalities with no solution or all real numbers as solutions. 1. 2. 3. Is there a solution for |x – 1| < -2? Why or why not? For C < 0, n >0: Is there a solution to 3|x-1| > -6? What numbers will make |x – 1| > -2 true? What is the solution? For C < 0, n < 0: Ask students to solve -3|x – 1| < -6. What is the simplified form of this equation? Which compound inequality would you use to solve? Why is it not advisable to use the distributive property when solving absolute value inequalities? What is the definition of absolute value? Can distance have a negative value? Can the absolute value of a number be negative? How many numbers have an absolute value less than 0? …less than -2? How many numbers have an absolute value greater than zero? Greater than -1? Graphing Inequalities Overview of Objectives, students should be able to: Main Overarching Questions: 1. Graph (solve) a linear inequality in two variables. 2. Graph (solve) a system of linear equalities in two variables. 2. What is a boundary line? 1. How do you solve and graph a linear inequality in two variables? 3. How do you graph a system of linear inequalities in two variables? Objectives: Activities and Questions to ask students: • 1. Background knowledge: How do you graph y = 2x + 1? Graph the equation. 2. Ask students to name a point that would make y > 2x + 1. Shade the side of the line where the point lies. 3. Graph the line again on a new grid and ask students to name a point that makes y < 2x + 1. Shade the side where the point lies. 4. Define boundary line and the difference between dashed and solid boundary lines. 5. What is the difference in the solution to y > 2x + 1 and y > or = 2x + 1? Graph (solve) a linear inequality in two variables. The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. • 6. Background knowledge. Review graphing a system of linear equations. What is the solution to a system with intersecting lines? 7. Ask students to graph two inequalities that intersect on the same grid. What points do the two graphs have in common? How many? How does the graph show this? 8. Ask student to graph two inequalities on the same grid that do not overlap. What points do the inequalities have in common? What is the solution to this system? Graph (solve) a system of linear equalities in two variables. Problem Solving Overview of Objectives, students should be able to: 1. Use mathematical models involving linear inequalities. 2. Solve problems using absolute value inequalities. 3. Solve applied problems using linear inequalities. Main Overarching Questions: 1. Objectives: Activities and Questions to ask students: • Discuss the steps in problem solving with students. Review words that translate to inequalities such as exceeds, is no more than, at least, etc. • • • • Use mathematical models involving linear inequalities. Solve problems using absolute value inequalities. Solve applied problems using linear inequalities. The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government.