1. Chapter 2 - Blue Mrs. Kane

2. Warm-Up

3. Math in History The concept of writing all numbers by using only ten different symbols appears to have originated in India. Here are some of the symbols that were used in the Brahmi system in India until around 400 A.D.

4. Math in History Notice that there is no symbol for 0. That concept was not yet devised. Also notice that the symbol for 2 and 3 are related to our modern symbols for 2 and 3.

5. Math in History By comparing the Brahmi symbols to another culture’s symbols (such as the Chinese), you can see that our modern symbols are more closely related to the ancient Brahmi symbols.

6. What You Learned Before Evaluating Expressions – Order of Operations 1)Grouping 2)Exponents 3)Multiply and Divide – left to right 4)Add and Subtract – left to right

7. What You Learned Before Graphing on a coordinate grid – Order pairs (x, y) – x-axis is the horizontal line – y-axis is the vertical line – Go right to left and then up or down

8. 2.1 Graphing Linear Equations - Activity Coordinate BINGO – Plot ten ordered pairs, where the x- and y- coordinates are integers between -4 and 4. – If I call one of your pairs put an X there. – The goal is to be the first person with three X’s. – Remember: (x, y): x is the horizontal direction and y is the vertical direction

9. 2.1 Graphing Linear Equations - Activity Are there ordered pairs that are not on lattice points, meaning the x- or y-coordinate is not an integer? Explain

10. 2.1 Graphing Linear Equations Vocabulary: – Linear equation: an equation whose graph is a line – Solution of a linear equation: the points on the line

11. 2.1 Graphing Linear Equations

12. Steps to graphing a linear equation using a table 1.Draw a t-table 2.Use at least three values for x 3.Solve the x values for y 4.Plot the ordered pairs 5.Draw a line through the points

13. 2.2 Slope of a Line - Activity How many of you have been on a roller coaster? What makes one roller coaster more thrilling than another?

14. 2.2 Slope of a Line - Activity

15. Can the change in x be negative? Can the change in y be negative?

16. 2.2 Slope of a Line Vocabulary: – Slope: the ratio of the change in y to the change in x – Rise: change in y – Run: change in x

17. 2.2 Slope of a Line Positive Slope: line rises left to right Negative Slope: line falls from left to right

18. 2.2 Slope of a Line

19. Key Idea: – Two lines in the same plane that do not intersect are parallel lines – Two lines with the same slope are parallel

20. 2.3 Graphing Linear Equations in Slope-Intercept Form (activity)

21. How do you graph an equation? How do you organize the points you need to plot? How many points do you need to your t-table?

22. 2.3 Graphing Linear Equations in Slope-Intercept Form How do you think taxi fares are determined? Why do some locations have flat fees?

23. 2.3 Graphing Linear Equations in Slope-Intercept Form Vocabulary: – x-intercept: where the line crosses the x-axis (it occurs when y = 0) – y-intercept: where the line crosses the y-axis (it occurs when x = 0)

24. 2.3 Graphing Linear Equations in Slope-Intercept Form

25. 2.4 Graphing Linear Equation in Standard Form (Activity) Four volunteers Raise your hand if the ordered pair is a solution to your equation (0, 1) (1, 3) (-1, 0) (2, 4) (-2, 3) (3, 7)

26. 2.4 Graphing Linear Equation in Standard Form (Activity) Plot the points that are solutions How many lines can pass through any two points? How many lines pass through the four solutions points?

27. 2.4 Graphing Linear Equation in Standard Form (Activity) Activity 1 – What does the x represent? – What does the y represent? – Can you sell 5 adult tickets? Why? – Can x = 1.5? Why? – What are the different numbers of the adult tickets that are possible to sell?

28. 2.4 Graphing Linear Equation in Standard Form How many pairs of numbers can you think of that add to 5? Did any of you include numbers that are not whole numbers? Name the x-coordinate in one of your ordered pairs. Who can name the y-coordinate for that pair? Plot on the coordinate plane. What do you think the equation of this line would be?

29. 2.4 Graphing Linear Equation in Standard Form Key Idea: – Standard Form of a Linear Equation ax + by = c, where a and b are not both zero

30. 2.4 Graphing Linear Equation in Standard Form Different Forms of Equations Slope-Intercepty = mx + b 1.Plot (0, b) 2.Use slope, m to plot the second point 3.Draw a line through the two points Horizontal Liney = c1.Draw a horizontal line through (0, c) Standard Formax + by = c 1.Find the y-intercept 2.Find the x-intercept 3.Plot the associated points. 4.Draw a line through the two points.

31. 2.5 Systems of Linear Equations (Activity) Have any of you visited a place where a monument was located in the middle of two streets? Monument Circle in Indianapolis General Palmer in Colorado Springs

32. 2.5 Systems of Linear Equations (Activity) System of Linear Equations: a set of two or more linear equations

33. 2.5 Systems of Linear Equations (Activity) Vocabulary Check: – Fixed cost – Variable cost – Revenue (income) Activity 2: – Why would a business want to know the break even point?

34. 2.5 Systems of Linear Equations Vocabulary: – System of Linear Equations: a set of two or more linear equations – Solution of a system of linear equation: an ordered pair that makes both equations true

35. 2.5 Systems of Linear Equations Methods of solving a system of linear equations: 1.Using a Table Make a table of values Find the x-value that gives you the same y-value Solution: (5, 0) x012345 y = x - 5-5-4-3-20 y = -x + 5543210

36. 2.5 Systems of Linear Equations Methods of solving a system of linear equations: 2. Using a graph Graph each equation Find the point of intersection Check your solution

37. 2.5 Systems of Linear Equations Methods of solving a system of linear equations: 3. Algebraically – Solve both equations for one of the variables – Set the equations equal to each other and solve – Substitute back into the original equation and solve for the other variable

38. 2.6 Special Systems of Linear Equations (Activity) Here are two intersecting lines. Are there other relationships that two lines can have?

39. 2.6 Special Systems of Linear Equations (Activity) How do you measure the vertical distance between two graphs?

40. 2.6 Special Systems of Linear Equations What is the general form of an equation in slope-intercept form? Distribute cards Match the equation to it’s slope and y-intercept.

41. 2.6 Special Systems of Linear Equations Review: – How do you graph equations in slope-intercept form?

42. 2.6 Special Systems of Linear Equations What are the different things that can happen when you are solving a system of equations? – One solution – line intersect – No solution – lines are parallel – same slope but different y-intercepts – Infinitely many solutions – lines and equations are the same

43. 2.7 Solving Equation by Graphing (Activity) What equation is being represented in this problem? How can you solve? =

44. 2.7 Solving Equation by Graphing Key Idea: – Solving equations using graphs 1.To solve the equation ax + b = cx + d, write two linear equations y = ax + b and y = cx + d 2.Graph the system of linear equations 3.Check your answer