Integrated Mathematics 8 - Past Paper PDF

INTEGRATED MATHEMATICS 8 Ms. Ana Lea G. Degorio MOST ESSENTIAL LEARNING COMPETENCIES The learner: illustrate the rectangular coordinate system and its uses. M8ALIe-1 Quadrant II A 𝒙- axis c Quadrant III B Definition 1.1 Ordere...

INTEGRATED MATHEMATICS 8 Ms. Ana Lea G. Degorio MOST ESSENTIAL LEARNING COMPETENCIES The learner: illustrate the rectangular coordinate system and its uses. M8ALIe-1 Quadrant II A 𝒙- axis c Quadrant III B Definition 1.1 Ordered pair refers to a set of two numbers used to locate a point in a coordinate plane. It is in the form (𝑥, 𝑦).  (−𝟐, 𝟑) abscissa ordinate 𝑥- coordinate 𝑦- coordinate (−𝟐, 𝟑) A (𝟑, 𝟎) c (−𝟒, −𝟒) B  INDIVIDUAL LEARNING TASK 1 Instruction: Describe the location of each objects in the plane by completing the table below.  INDIVIDUAL LEARNING TASK 2 INDIVIDUAL LEARNING TASK Goal: The student plot the points in the cartesian plane. Materials: graphing paper, pencil, ruler Instruction: 1. Draw Cartesian plane that has coordinates ranging from 1 to 10 on both the 𝑥-axis (horizontal) and the 𝑦-axis (vertical). 2. Label the axes clearly to ensure you can easily find specific coordinates. 3. Plot the following ordered pairs. Label the points after plotting. 𝑨 𝟗, 𝟏 𝑩 𝟎, 𝟖 𝑪 −𝟗, 𝟏 𝑫 𝟔, −𝟖 𝑬 (−𝟔, −𝟖) Rubric: Cartesian Plane Accuracy Coordinates are labeled correctly from 1-10 on both axes; grid is clear and precise. Coordinate Indication All ordered pairs plotted accurately and labeled correctly. COLLABORATIVE LEARNING TASK 1 COLLABORATIVE LEARNING TASK Goal: The students create 2D community map. Materials: Illustration board ¼ size, printed buildings, ruler, scissors, colors, and glue. Hospital Church Park School Police Station Market Convenient Store Fire Station City/Barangay Hall Instruction: 1. Draw a Cartesian plane. 2. Label the axes clearly to ensure you can easily find specific coordinates. 3. Prepare and cut out printed images of different buildings (e.g., houses, schools, shops). 4. Decide where to place each building. Instruction: 5. Once you have chosen the locations for your buildings, use glue or tape to paste the cutouts onto the Cartesian plane. 6. Use the 𝑥-axis and 𝑦-axis to determine the ordered pairs (coordinates) for each building. 7. Write down the ordered pair for each building directly on the map, next to or below the building. Instruction: 8. Add additional elements like roads, trees, and parks to make your map more realistic and visually appealing. 9. Use colors, markers, and other decorations to enhance the final look of your community map. Rubric: Cartesian Plane Accuracy- 10 points Coordinates are labeled correctly on both axes; grid is clear and precise. Coordinate Indication- 15 points All buildings have their ordered pairs accurately labeled next to them. Creativity and Design- 15 points Map is highly creative and well-decorated with clear details, colors, and added features like roads, trees, etc. Neatness and Organization- 10 points Map is extremely neat and organized; buildings and coordinates are clearly visible. Guided Questions for Presentation 1. Briefly introduce your community map and explain its overall design. MOST ESSENTIAL LEARNING COMPETENCIES ❑ Illustrates linear equations in two variables. M8ALIe-3 ❑ Illustrates and finds the slope of a line given two points, equation, and graph.  Definition 2.1 Linear Equation in Two Variables is an equation with two variables, each of which has an exponent of one. Determine whether the following equation is a linear equation in two variables. 4𝑥 − 5𝑦 = 100 1 3𝑏 − 4𝑦 2 = 20 4𝑧 −4 = 20 + 3𝑦 𝑐 = −10 + 2𝑎 𝑥 = 13 14 = 4𝑦 + z Linear Equations in Two Variables 4𝑥 − 5𝑦 = 100 𝑐 = −10 + 2𝑎 14 = 4𝑦 + z  Definition 2.2 The solution of the equation is an ordered pair that satisfies the given equation.  Example 1 Determine if the ordered pair (−2,−1) satisfies the equation 𝑥 − 2𝑦 = 0. Solution: 𝑥 = −2, and 𝑦 = −1 −2 − 2 −1 = 0 −2 + 2 = 0 0=0 Therefore, (−2, −1) is the solution of the equation.  Example 2 Determine if the ordered pair (5, 0) satisfies the equation 2𝑥 − 5𝑦 = 10. Solution: 𝑥 = 5, and 𝑦 = 0 2 5 − 5 0 = 10 10 − 0 = 10 10 = 10 Therefore, (5,0) is the solution of the equation.  Example 3 Verify if the ordered pair (1, −1) satisfies the equation −3𝑥 + 𝑦 = 7. Solution: 𝑥 = 1, and 𝑦 = −1 −3 1 + −1 = 7 −3 − 1 = 7 −4 ≠ 7 Therefore, (1, −1) is not the solution of the equation.  Example 4 Which of the two ordered pair is the solution of the linear equation 3𝑥 − 𝑦 = 12? A. 9,2 B. (5, 3) 𝟑 𝟗 − 𝟐 = 𝟏𝟐 𝟑 𝟓 − 𝟑 = 𝟏𝟐 𝟐𝟕 − 𝟐 = 𝟏𝟐 𝟏𝟓 − 𝟑 = 𝟏𝟐 𝟐𝟓 ≠ 𝟏𝟐 𝟏𝟐 = 𝟏𝟐  The graph of the linear equation in two variables is a line.  Forms of Linear Equations A. Slope- intercept Form 𝑦 = 𝑚𝑥 + 𝑏 Examples: 𝑦 = −3𝑥 − 13 𝑦 = 𝑥 + 14  Forms of Linear Equations B. Standard Form 𝑎𝑥 + 𝑏𝑦 = 𝑐 Examples: 3𝑥 − 4𝑦 = 7 𝑥 + 2𝑦 = −10  Forms of Linear Equations C. General Form 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 Examples: 𝑥 + 𝑦 − 20 = 0 3𝑎 − 4𝑏 + 10 = 0  INDIVIDUAL LEARNING ACTIVITY Which of the two ordered pair is the solution of the linear equation 3𝑥 + 5𝑦 = 26? A. 2,6 B. (2, 4) 𝟑 𝟐 + 𝟓(𝟔) = 𝟐𝟔 𝟑 𝟐 + 𝟓(𝟒) = 𝟐𝟔 𝟔 + 𝟑𝟎 = 𝟐𝟔 𝟔 + 𝟐𝟎 = 𝟐𝟔 𝟑𝟔 ≠ 𝟐𝟔 𝟐𝟔 = 𝟐𝟔  Definition 3.1 Slope is a ratio of the change in vertical distance (rise) to the change in horizontal distance (run); usually denoted by the variable 𝑚.  Definition 3.1 ▪ The steepness, incline, or grade of a line is measured by the absolute value of the slope. ▪ The higher the value of slope, the steeper the line is.  Methods of Finding the Slope A. Given the Graphs or Illustrations We use the formula: 𝑟𝑖𝑠𝑒 𝑚= 𝑟𝑢𝑛 Example 1: The rise is 4 steps. The run is 2 steps going to the right direction. Thus, 𝑟𝑖𝑠𝑒 4 𝑚= = =2 𝑟𝑢𝑛 2 Example 2: The rise is 5 steps. The run is 4 steps going to the left direction. Thus, 𝑟𝑖𝑠𝑒 5 5 𝑚= = =− 𝑟𝑢𝑛 −4 4  Methods of Finding the Slope B. Given the Two Points We use the formula: 𝑟𝑖𝑠𝑒 𝑦2 − 𝑦1 𝑚= = 𝑟𝑢𝑛 𝑥2 − 𝑥1 Example 1: Find the slope of the line that passes through the points (5,−4) and (1, 5). Step 1: Identify the values of the variables. 𝑥1 = 5 𝑦1 = −4 𝑥2 = 1 𝑦2 = 5 Step 2: Substitute the values to the formula. 𝑦2 − 𝑦1 𝑚= 𝑥2 − 𝑥1 5 − (−4) 1−5 Example 1: Find the slope of the line that passes through the points (5,−4) and (1, 5). Step 3: Simplify the expression into lowest term. 5 − (−4) 1−5 5+4 −4 9 𝑚=− 4 Example 2: Determine the slope of the line that passes through the points (4,−5) and (−2,−8). Step 1: Identify the values of the variables. 𝑥1 = 4 𝑦1 = −5 𝑥2 = −2 𝑦2 = −8 Step 2: Substitute the values to the formula. 𝑦2 − 𝑦1 𝑚= 𝑥2 − 𝑥1 −8 − (−5) −2 − 4 Example 2: Determine the slope of the line that passes through the points (4,−5) and (−2,−8). Step 3: Simplify the expression into lowest term. −8 − (−5) −2 − 4 −8 + 5 −6 −3 1 𝑚= = −6 2  INDIVIDUAL LEARNING ACTIVITY Solve the slope of the line that passes through the points (−3,9) and (4,2).  Methods of Finding the Slope C. Given the Equations If the equation is in the standard form, use the 𝐴 formula −. 𝐵  Determine the slope of the following linear equations. 4 4 1. 4𝑥 − 5𝑦 = 100 𝑚 = − 𝑚= −5 5 2 2. 2𝑎 + 𝑐 = −10 𝑚 = − 𝑚 = −2 1 4 3. 4𝑦 + 3z = 14 𝑚=− 3  Methods of Finding the Slope C. Given the Equations If the equation is in the slope- intercept form, determine 𝑦 = 𝑚𝑥 + 𝑏. Determine the slope of the following linear equations. 1. 𝑦 = 2𝑥 + 5 𝑚=2 3 3 2. 𝑦 = − 𝑥 −7 𝑚=− 4 4 3. 𝑦 = 𝑥 + 1 𝑚=1  Slope of the Lines The two lines are parallel each other if and only if their slopes are the same.  Slope of the Lines The two lines are perpendicular to one another if and only if their slopes are negative reciprocal.  MOST ESSENTIAL LEARNING COMPETENCY The learner: writes the linear equation 𝑎𝑥 + 𝑏𝑦 = 𝑐 in the form 𝑦 = 𝑚𝑥 + 𝑏 and vice versa.  REVIEW Standard Form of a Linear Equation The standard form of a linear equation in two variables 𝑥 and 𝑦 is an equation in the form 𝒂𝒙 + 𝒃𝒚 = 𝒄: where 𝑎, 𝑏, and 𝑐 are real numbers 𝑎 and 𝑏 are not both equal to 0 by convention, 𝑎 should also be nonnegative.  Forms of Linear Equations B. Standard Form 𝑎𝑥 + 𝑏𝑦 = 𝑐 Examples: 3𝑥 − 4𝑦 = 7 𝑥 + 2𝑦 = −10  REVIEW Slope-Intercept Form of a Linear Equation The slope-intercept form of a linear equation in two variables 𝑥 and 𝑦 is an equation of the form 𝒚 = 𝒎𝒙 + 𝒃 where 𝑚 is the slope and 𝑏 is the 𝑦-intercept.  Forms of Linear Equations A. Slope- intercept Form 𝑦 = 𝑚𝑥 + 𝑏 Examples: 𝑦 = −3𝑥 − 13 𝑦 = 𝑥 + 14 AXIOM Addition Property of Equality If 𝑎 = 𝑏, then 𝑎 + 𝑐 = 𝑏 + 𝑐. Example: 𝑥+5=3 𝑥+5−5=3−5 𝑥 = −2 TRANSPOSE Example: 𝑥+5=3 𝑥 =3−5 𝑥 = −2  TRANSFORMING Standard form to Slope- intercept form  Example 1 Transform 𝑥 + 𝑦 = 11 to slope- intercept form. Solution: 𝑥 + 𝑦 = 11 𝑦 = −𝑥 + 11  Example 2 Transform 7𝑥 + 𝑦 = −12 to slope- intercept form. Solution: 7𝑥 + 𝑦 = −12 𝑦 = −7𝑥 − 12  Example 3 Transform 10𝑥 − 𝑦 = 5 to slope- intercept form. Solution: 10𝑥 − 𝑦 = 5 −𝑦 = −10𝑥 + 5 𝑦 = 10𝑥 − 5  Example 4 Transform 4𝑥 + 2𝑦 = 14 to slope- intercept form. Solution: 4𝑥 + 2𝑦 = 14 2𝑦 = −4𝑥 + 14 2𝑦 −4𝑥 14 = + 2 2 2 𝑦 = −2𝑥 + 7  Example 5 Transform 3𝑥 − 2𝑦 = 5 to slope- intercept form. Solution: 3𝑥 − 2𝑦 = 5 −2𝑦 = −3𝑥 + 5 −2𝑦 −3𝑥 5 = + −2 −2 −2 3 5 𝑦= 𝑥− 2 2  TRANSFORMING Slope- intercept form to Standard form  Example 1 Transform 𝑦 = −𝑥 + 11 to standard form. Solution: 𝑦 = −𝑥 + 11 𝑥 + 𝑦 = 11  Example 2 Transform 𝑦 = −2𝑥 + 7 to standard form. Solution: 𝑦 = −2𝑥 + 7 2𝑥 + 𝑦 = 7  Example 3 Transform 𝑦 = 7𝑥 − 12 to standard form. Solution: 𝑦 = 7𝑥 − 12 −7𝑥 + 𝑦 = −12 7𝑥 − 𝑦 = 12  Example 4 3 5 Transform 𝑦 = 𝑥 − to standard 2 2 form. Solution: 3 5 2 𝑦= 𝑥− 2 2 2 2𝑦 = 3𝑥 − 5 −3𝑥 + 2𝑦 = −5 3𝑥 − 2𝑦 = 5  Example 5 5 7 Transform 𝑦 = − 𝑥 + to standard 3 4 form. Solution: 5 7 12 𝑦 = − 𝑥 + 12 3 4 12𝑦 = −20𝑥 + 21 20𝑥 + 12𝑦 = 21 COLLABORATIVE LEARNING ACTIVITY 1  2 3 Is 4𝑥 − 10𝑦 = 15 equivalent to 𝑦 = 𝑥 + ? 5 2 Show your solution. 1. 4𝑥 − 10𝑦 = 15 Slope-intercept Form: 2 3 2. 𝑦 = 𝑥 + 5 2 Standard Form: Conclusion:  MOST ESSENTIAL LEARNING COMPETENCY The learner: graphs a linear equation given (a) any two points; (b) the –𝑥 and –𝑦 intercepts; (c) the slope and a point on the line; describes the graph of a linear equation in terms of its intercepts and slope.  The graph of the linear equation in two variables is a line. GRAPHING Using the intercepts  𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 Definition Point where a Point where a line crosses the line crosses the x- axis y- axis Coordinates (𝑥, 0) (0, 𝑦) How to find Set 𝑦 = 0 and Set x= 0 and solve for 𝑥 solve for 𝑦 Example 1: Graph the equation 4𝑥 − 3𝑦 = 18. 𝒙 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 𝒚 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 Set 𝑦 = 0 and solve for 𝑥 Set x= 0 and solve for 𝑦 4𝑥 − 3 0 = 18 4 0 − 3𝑦 = 18 4𝑥 = 18 −3𝑦 = 18 4𝑥 18 −3𝑦 18 = = 4 4 −3 −3 18 𝑦 = −6 𝑥= = 4.5 4 (0, −6) (4.5,0) GRAPHING Using the slope and intercept COLLABORATIVE LEARNING ACTIVITY 1 Find your partner. Goal: The student will graph linear equation using intercepts, and slope and a point. Material: Graphing Paper, Ruler, Pencil Instruction: Create cartesian plane in your graphing paper. Graph the following equations. Show your solutions in finding the intercepts. 1. Graph the equation using intercepts. 5𝑥 + 10𝑦 = 20 2. Graph the equation using the slope and a point. 4 𝑦 =− 𝑥−9 3 INDIVIDUAL LEARNING ACTIVITY 1 Answer the given quiz in the Quipper. MOST ESSENTIAL LEARNING COMPETENCIES The learner: finds the equation of a line given (a) two points; (b) the slope and a point; (c) the slope and its intercepts. Method 1: Two- Point Form If the graph of a linear equation passes through the points (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ), then its equation is 𝒚𝟐 − 𝒚𝟏 𝒚 − 𝒚𝟏 = (𝒙 − 𝒙𝟏 ) 𝒙𝟐 − 𝒙𝟏 Example 1: Find the equation of a line that passes through the points (5,6) and (3,2) as shown on the graph. Example 1: Find the equation of a line that passes through the points (5,6) and (3,2) as shown on the graph. Solution: 𝑦2 − 𝑦1 𝑦 − 𝑦1 = (𝑥 − 𝑥1 ) 𝑥2 − 𝑥1 𝒙𝟏 = 𝟓 𝒚𝟏 = 𝟔 𝒙𝟐 = 𝟑 𝒚𝟐 = 𝟐 2−6 𝑦−6= (𝑥 − 5) 3−5 −4 𝑦−6= (𝑥 − 5) −2 𝑦 − 6 = 2(𝑥 − 5) Example 1: Find the equation of a line that passes through the points (5,6) and (3,2) as shown on the graph. Solution: 𝑦 − 6 = 2(𝑥 − 5) 𝑦 − 6 = 2𝑥 − 10 𝑦 = 2𝑥 − 10 + 6 𝒚 = 𝟐𝒙 − 𝟒 or 𝟐𝒙 − 𝒚 = −𝟒 Example 2: Find the equation given the two points (-2, 3) and (1, 2) which passes on the line. Solution: 𝑦2 − 𝑦1 𝑦 − 𝑦1 = (𝑥 − 𝑥1 ) 𝑥2 − 𝑥1 𝒙𝟏 = −𝟐 𝒚𝟏 = 𝟑 𝒙𝟐 = 𝟏 𝒚𝟐 = 𝟐 2−3 𝑦−3= (𝑥 − (−2)) 1 − −2 −1 𝑦−3= (𝑥 + 2) 3 Example 2: Find the equation given the two points (-2, 3) and (1, 2) which passes on the line. Solution: −1 𝑦−3= (𝑥 + 2) 3 1 2 𝑦−3=− 𝑥− 3 3 1 2 3 𝑦−3=− 𝑥− 3 3 3 3𝑦 − 9 = −𝑥 − 2 Example 2: Find the equation given the two points (-2, 3) and (1, 2) which passes on the line. Solution: 3𝑦 − 9 = −𝑥 − 2 𝑥 + 3𝑦 = −2 + 9 𝟏 𝟕 𝒙 + 𝟑𝒚 = 𝟕 or 𝒚 = − 𝒙 + 𝟑 𝟑 Example 3: Find the equation of a line that passes through the points (2,3) and (4, −2). Solution: Method 2: Point Slope Form If the graph of a linear equation has a slope and passes through (𝑥1 , 𝑦1 ) then its equation is 𝒚 − 𝒚𝟏 = 𝒎(𝒙 − 𝒙𝟏 ) Example 1: The slope of the line is −2 which passes through point (2, 3). 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) 𝒎 = −𝟐 𝒙𝟏 = 𝟐 𝒚𝟏 = 𝟑 𝑦 − 3 = −2(𝑥 − 2) 𝑦 − 3 = −2𝑥 + 4 𝑦 = −2𝑥 + 4 + 3 𝒚 = −𝟐𝒙 + 𝟕 or 𝟐𝒙 + 𝒚 = 𝟕 Example 2: Write the equation of a line whose graph has slope of 4 and a point (3, −3). 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) 𝒎=𝟒 𝒙𝟏 = 𝟑 𝒚𝟏 = −𝟑 𝑦 − (−3) = 4(𝑥 − 3) 𝑦 + 3 = 4𝑥 − 12 𝑦 = 4𝑥 − 12 − 3 𝒚 = 𝟒𝒙 − 𝟏𝟓 or 𝟒𝒙 − 𝒚 = 𝟏𝟓 Example 3: Write an equation of the line in slope- intercept form given the line has a slope -3 and passes through the point (2,1). Solution:  COLLABORATIVE LEARNING ACTIVITY Instruction: The activity can be done in group (3 members). Solve the equation of the following given. Write your answers and solutions in ½ crosswise. 1. Find the equation of a line that has point of (5, −2) and (1,0) 2. Find the equation of a line that has a slope of -7 and a point of (−13, −10). MOST ESSENTIAL LEARNING COMPETENCIES The learner: solves problems involving linear equations in two variables. Problem 1: Mario’s city was built similar to a rectangular coordinate plane. Using Mario’s house as origin, the church is located at the point (5, 3). The mall is located at the point (5, −6). If each unit represents 100 meters, how far is the church from the mall? Problem 2: Two branches of a famous mall, 𝐴 and 𝐵 are connected by a train that has a station at point (4, −2) of the coordinate plane. Mall A lies on the point (2, −5) and Mall B lies on the point (3, −4). Which of the two malls is closer to the train station? Problem 3: Mr. San Pedro paid ₱360 for 6 adult’s tickets and 7 children’s tickets for the school play. Write a linear equation for the given situation with the cost of the tickets unknown. Let be the cost of an adult’s ticket and be the cost of a child’s ticket. Problem 4: As Marina eats frequently in her favorite restaurant, she noticed that the time it takes the food to be served depends on how many people she is ordering for. If she orders for two people, their food gets served in about 10 minutes. If she goes with her family, the six of them will have to wait for 20 minutes to get served. What is the rate at which people are served their food? Problem 5: You want to buy a present for your parent’s wedding anniversary and you decided to save ₱200 a month from your allowance. You have an initial savings of ₱400. How much will you save in 3 months? Problem 6: Eduardo receives 5% commission for his sales and additional allowances of ₱10 000 every month. If he has ₱68 000 sales, how much would he earn for the month? Problem 7: A car rental company charges ₱4000 a day and then ₱50 for every kilometer that the car travels. If Luisa rented the car for one day and travelled 15 kilometers, how much would she pay the company. Problem 8: Nicole saved 20-peso bills and 50-peso bills for a month. She counted the 20- peso bills and found out that she has saved 30 pieces of the said bill. If she hoped to save a total amount of ₱1200, how many 50-peso bills should she have saved? Problem 9: Ray and Angel work in a textile company. Ray’s rate is ₱150 per hour while Angel’s rate is ₱200 per hour. If their combined income in a week is ₱8 500 and if Angel worked for 20 hours that week, how many hours did Ray work that week?  WRAP UP How would you illustrate Cartesian Coordinate Plane? How would you illustrate linear equation in two variables? How would you illustrate and find the slope of a line?  WRAP UP How would you transform linear equations into different forms? How would you graph a linear equation in two variables? How would you solve a linear equation in two variables given a slope and intercepts?