Lesson 7.1
Cartesian Coordinate System--Recisited
(Textbook pages 514-519, 2 sessions)
I. Mathematical Concepts and Skills
A. The Cartesian coordinate system is consist of two coplanar perpendicular number lines called axes. The point of intersection of the horizontal line (or the x-axis) or the vertical lines(or the y axis) is called the origin. These axes divide the plane into four regions which are called quadrants.
B. There is on-to-one correspondence between the points on the Cartesian plane and the ordered pairs of real numbers.
C. An ordered pair is a set of two well-ordered real numbers called coordinates.
D. The coordinated of a point on the Cartesian plane is determined by projecting the points on both axis.
II. Objectives
A. Name the parts of a Cartesian coordinate plane
B. Describe the distance of a point from two given perpendicular lines
C. Plot the point corresponding to a given pair of coordinates
D. Give the coordinates of a given point on the coordinate plane
E. Determine the location of a given point on Cartesian coordinate plane
III. Values Integration
Appreciate the contributions of great men of humanity
IV. Materials
Philippine and world map
Graphing board and colored chalk
Grid papers
Picture of Rene Descartes
V. Instructional Strategies
A. Exposition
B. Guided Discovery
C. Consolidation and Practice
VI. Procedure
Session I
A. Preliminary Activities
1. Check the assignment
2. Provide this motivation:
A. Do you know that elephant can dance?
B. Do you know how elephants dance?
Teach the class the elephant dance. Ask them to move five steps to the left, six steps backward, four steps to the right and seven steps forward. This exercise will enhance their sense of direction. Use music if available.
What directions connote positive numbers? Negative numbers?
B. Lesson Proper
1. Draw two perpendicular lines on the boars, one horizontal and the other vertical. Put some points anywhere on the plane. Label the points. Ask the students to describe the location of each point as to whether it is above, below, to the right, or to the left of any of the horizontal line or below?
2. Make a thorough exposition of the Cartesian coordinate system. Through guided questions, let the students identify important parts of the Cartesian plane such as the x-and -axes, the origin, and the quadrants.
3. Briefly recall the concept pf one-to-one correspondence. Let the students realize that every point on the Cartesian plane has a corresponding ordered pair of real numbers.
4. Discuss how to plot on the Cartesian plane. Use Example 1 on page 515 of the textbook. Give more examples.
5. Discuss how to name the coordinates of the point on the Cartesian coordinate plane . Use Example 2 on page 516. If necessary, give other examples.
6. Tell the class that the concept locating points on the Cartesian plane was introduced by Rene Descartes. He used this concept in locating and describing places on the map, which is one of his important contributions to humanity. Show a picture of Descartes and briefly relate important facts about him. Give examples of locating a point on the map using the latitude and longitude, and vise versa.
7. Inculcate the importance of gratitude to those who made great contributions to humanity like Descartes.
8. Ask the class to consolidate the discussion. Let them answer Mental Mathematics Exercise 7.1 on page 517 of the textbook.
9. Tell them to work on Written Mathematics of Exercise 7.1 (numbers 1-10) on page 517 of the textbook.
C. Assignment
Work on the Written Mathematics of Exercise 7.1 (numbers 11-20) on page 518 of the textbook.
Session 2
A. Preliminary Activities
1. Check the assignment.
2. Give these reinforcement exercises.
Plot each point on the Cartesian coordinate plane.
A. (5 , -8) B. ( -1, 6)
C. (3 , 7) D. (-4 , -2)
E. (O , -8) F. (9 , 0)
Name the coordinate of each point.
A. A
B. B
C. C
D. D
E. E
F. F
B. Follow-up Lesson
1. Make a brief recall of the quadrants of the Cartesian coordinate plane. Introduce the terms abscissa and ordinate. Tell the class that the abscissa are the x-coordinate and the ordinates are the y-coordinates.
2. Ask them to give the abscissa and ordinates of the following ordered pairs.
(2 , 7) (-8 , 6) (11 , 0) (4, -10) ( 0, 9)
3. Divide the class in groups of 5 and let each group perform the activity below. Give each group a grid paper.
A. Plot each of the following points:
A ( 2, 7) B ( 3-, 6)
C (5 , -4) D ( -3, -8)
E ( 0, 4) F ( -6, -2)
G ( -10, 0) H ( 9, 1)
I (-8 , 1) J (1, -5)
B. Complete the table below.
Points Sign of Abscissa Sign of Ordinate Locatio
A ( 2, 7)
B ( -3, 6)
C (5 , -4)
D ( -1, -8)
E ( 0, 4)
F ( -6, -2)
G ( -10, 0)
H ( 9, 1)
I (-8 , 1)
J (1, -5)
4. Discuss the results of the activity by asking them to observe the table. Ask them to draw conclusions. Focus their attention on the signs of the coordinates and the location of the point.
5. Let the students answer Written Mathematics of Exercise 7.1 (numbers 11-20) on page 518 of the textbook.
6. Ask them to work in pairs in solving Written Mathematics of Exercise 7.1 (numbers 21-23) on page 519 of the textbook.
C. Checking for Understanding
1. Plot the following :
A. M ( 6, -2) B. N (-4, -8) C. P (0, -5)
2. Name the coordinates of the points on the Cartesian coordinate plane.
3. Determine the location of each point without plotting them.
A. (-68, 72) B. (-91, -108)
C. (168, -76) D. y=0, x > 0
E. x <0, y >0 F. x > 0, y > 0
G. x= 0, y < 0
H. positive abscissa, negative ordinate
I. Abscissa and ordinate having the same sign
J. Abscissa and ordinate are both unsigned
For fast learners, ask them to solve the Challenge Problems of Exercise 7.1 on page 519 of the textbook.
D. Assignment
Work on Written Mathematics oof Exercise 7.1 (numbers 24-28) on page 519 of the textbook.
Lesson 7.2
The Slope of a Line
(Textbook pages 520-531, 2 sessions)
I. Mathematical Concepts and Skills
A. The slope of a nonvertical line that passes through P₁(x₁,y₁) and P₂(x₂, y₂) is m= 
or m= 
.
or m= .B. A line with positive slope is pointing upward to the right while a line with negative slope is pointing upward to the left. A horizontal line has zero slope and for vertical lines, the slope is undefined.
C. The slope m of a nonvertical segment is equal to the slope of the line containing it.
D. Graphically, the slope of a line is determined by getting the ratio of the change in y to the change in x.
II. Objectives
A. Define the slope of a line
B. Find the slope of a line given two of its points
C. Find the slope of a segment
D. Determine through the graph whether a line has a positive, negative, or zero slope
E. Determine graphically the slope of the line
F. Graph a line given a point on a line and its slope.
Inculcate the value if perseverance
IV. Materials
graphing board, colored chalk, straight edge
grid papers
prepared illustrations of lines drawn on Cartesian planes
pictures of three boys climbing mountain with different degrees of steepness.
p. 149
V. Instructional Strategies
A. Exposition
B. Consolidation and Practice
VI. Procedure
Session I
A. Preliminary Activities
1. Check the assignment
2. Motivation
a. Ask: “Who among you had experienced mountain climbing?”
b. Show the pictures of three boys climbing a mountain. Ask them to observe the position of the three boys.
c. Assuming the three boys start climbing at the same time, who do you think could be the first to reach the top, disregarding other factors.
d. Students must realize that the pictures shown are mere representations of steepness of lines.
3. Discuss the importance of perseverance
B. Lesson Proper
1. Draw three lines on the board similar to those on page 520 of the textbook. Use these illustrations to compare the steepness between lines. Tell the students that the numerical value assigned to the steepness of the line is referred to as slope.
2. Tell the students that the slope can be computed using the formula is m= or m=
or m= The formula suggests the need to determine the two points on a line so as to get the slope. Discuss Examples 1 and 2 on pages 522-524 of the textbook.
3. Ask the class why the slope of a line is represented by m.
4. Discuss briefly the slope of a segment. Use Examples 3 and 4 on pages 524-525 of the textbook.
5. Ask the class to consolidate the discussion, then let them answer Written Mathematics of Exercise 7.2 (number 1—6) on pages 530-531 of the textbook.
Page 150
C. Assignment
Work on Written Mathematics of Exercise 7.2 (numbers 7-16) on page 530 of the textbook.
Session 2
A. Preliminary Activities
1. Check the assignment
2. Give these reinforcement exercises.
Find the slope of the line that passes through each pair of points.
a. (-2, 0). (6)3 b. (3,-10)
c. (5, -1), (4,5) d. (3.6)(3,10)
Plot each pair of points in the exercises above them graph the lines that pass through each pair.
3. Tell the students to observe each graph and their corresponding slope.
How can you relate the slope of a line to its graph?
B. Follow-up Lesson
1. Use the exercises above to make a throughout exposition of the graphs of lines with positive, negative, zero, or undefined slopes.
2. Post on the board the prepared illustration similar to the illustration on page 526 of textbook
3. Discuss how to graph a line given a point on a line and its slope. Use example on page 557. Give other examples.
4. Ask them to stand up and teach them to use their hands and feet to represent lines with positive, negative, zero, and undefined slopes.
Page 151
C. Checking for Understanding
Find the slope of the line that passes through each pair of points.
1. (3,-4),(-2,) 2. (-1, -5), (1, 3)
3. (1, 3), (7, 12) 4. (4, -2), (7, 3)
5. (5, -1), (-3, 2) 6. (-3, 6), (9, -18)
Find the value of the unknown so that the line that passes through the given pair of points will have the given slope.
7. (2, 1), (4, y); m=1/2 8. (x, 1), (2, -2); m=-3/5
9-10. Determine if the points A(-4, -1), B(-2, 1), and C(2, 5) are collinear.
For the fast learners, ask them to solve the Challenge Problems on page 531.
D. Assignment
Work on Written Mathematics of Exercise 7.2 (numbers 17-22) on page 531 of the textbook.
Lesson 7.3
Linear Equations
(Textbook pages 532-538, 2 sessions)
I. Mathematical Concepts and Skills
A. The standard form of the equation of a line is Ax+ By= C, where A,B,C are real numbers.
B. If the points P₁(x₁, y₁) and P₂(x₂, y₂) are on the line, then its equation may be written in the form
y₂-y₁= y₂-y₁/x ₂-x₁(x₂-x₁)
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C. If m is the slope of a line and b is its y-intercept, then its equation can be written in the form y = mx+b.
D. If P₁(x₁, y₁) is a point on a line whose slope is m, then its equation can be written in the form
y₂-y₁= m(x-x₁)
II. Objectives
A. Derive the different forms of the equation of a line
B. Rewrite a linear equation of the form Ax + By = C in the form y=mx+b, and vice versa.
C. Obtain the equation of a line given the following:
1. two points
2. slope and one point
3. slope and y-intercept and its graph
Include all the value of obeying laws
IV. Materials
Graphing board, colored chalk
Grid papers
Acetate or prepared board work for generalizations and exercises
V. Instruction Strategies
A. Use of Puzzles (jumbled words)
B. Discussion
C. Consolidation and Practice
VI. Procedure
Session 1
A. Preliminary Activities
1. Check the assignment.
2. Provide these puzzles
Rearrange the scrambled letters to form a word. A description of the word on the right serves as hint.
a. POLES- ratio of the “rise” to the “run.”
b. TINTCREPE- the intersection of the graph and an axis.
c. TIPON- part of a graph
Page 153
3. Provide a quick review of the different properties of equality.
4. Inculcate the importance of obeying laws.
B. Lesson Proper
1. Lead the students in recalling the standard form of the linear equation.
Discuss Example 1 on page 532
2. Use these additional exercises on transferring equations in standard form. Use flash cards.
a. y= -3x +4 b. 7x+ 2y- 3= o
c. x= 2y-3 d. 2x-y/ 2= 1
e. 4x-2 = 3y f. 1= 3x/2 – y
g. ½ x- 1/3= 4/5 y
3. Answer Mental Mathematics of Exercise 7.3 (numbers 1-10) on page 537 of the textbook.
4. Present Examples 2 and 3 on pages 532- 534.
5. Make use of the standard equation of the line in deriving the equation of a line that passes through the given
point with the given slope.
a. (9, 0), m= 2 b. (2, 4), m= 1
c. (2, 4), m= -3 d. (6, 3), m= ½
e. (0, 4), m= -3/5 f. (-6, -4), m= 2/3
6. Encourage the students to graph the lines above.
C. Assignment
Work on Written Mathematics of Exercise 7.3 (numbers 1-10) on page 537 of the textbook.
Page 154
Session 2
A. Preliminary Activities
1. Check the assignment
2. Give these reinforcement exercises.
Find the equation of a line that passes through the given points with the given slope.
a. (4, 6), m= 1 b. (-2, 1), m= -2
c. (7, -4), m= -1/2 d. (-3, -8), m= -1
e. (1, 3), m=3 f. (4, -1/2), m= 4/3
B. Follow-up Lesson
1. Discuss how to find the equation of a line through two points given points. Present Example 4 on page 535. If
necessary, provide additional examples.
2. Make use of the standard form of the equation of the line in deriving the slope-intercept form of the equation of
the line.
3. Present Example 5 on page 536
4. Let the students answer Mental Mathematics of Exercise 7.3 (numbers 11- 20) on page 537 of the textbook.
5. Let the students work in pairs in solving these seatwork exercises. Obtain the equation of a line given the
following:
a. m=3, b= -7
b, m= -8/7, (4, -7)
c. (5, 2), (7, 6)
d. m= -1/2, b= 2
e. m= -1, b= 3
Page 155
Determine the equation of a line of the following graphs:  | |||
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f. g.
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C. Checking for Understanding
1. Obtain the equation of the line in slope- intercept form given the following slope and y-intercept of the line.
a. m= -1, b= -4 b. m=-1/2, b= 4 c. m=2, b=1/2
2. Determine the slope and y- intercept of each equation.
a. -4x + y= 9 b. x= 3+ 2y c. 12= 6x +3y 3. Determine the equation of each line in a standard form.
a. | |||
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4. Obtain the equation of a line in standard form given the following:
a. y- intercept: 7, slope: -2
b. x- intercept: -6, slope: 3
c. slope: -7/2, (3, -1)
d. (4, 0), (7, 2)
For fast learners, ask them to solve the Challenge Problems on page 538 of the textbook.
D. Assignment
Work on Written Mathematics of Exercise 7.3 (numbers 11-24) on page 538 of the textbook.
Page 159
Lesson 7.4
Parallel and Perpendicular Lines
(Textbook pages 539-547, 3 sessions)
I. Mathematical Concepts and Skills
A. If two nonvertical lines intersect, then their slopes are not equal.
B. If two nonvertical lines have different slopes, then the lines are intersecting.
C. If two nonvertical lines are parallel, then the slopes are equal.
D. If two lines which do not coincide have the same slope, then the lines are parallel,
E. If two nonvertical lines are perpendicular, then their slopes are negative reciprocals.
II. Objectives
A. Determine algebraically the point of intersection of two lines.
B. State and apply theorems on the slopes of intersecting, parallel and perpendicular lines.
Think logically
IV. Materials
Graphing board, colored chalk, straightedge
Grid papers
Visual aids on the theorems and corresponding proofs.
V. Instructional Strategies
A. Review
B. Exposition
C. Discussion
D. Consolidation and Practice
VI. Procedure
Session 1
A. Preliminary Activities
1, Check the assignment
Page 160
2. Motivation
a. To enhance the students’ logical thinking skills, let them try this activity:
“How can two identical squares be cut so the pieces can be rearranged to form bigger square?”
 | |
B. Lesson Proper
1. Recall how to determine the intersection of two lines algebraically. Use Examples 1-3 on pages 539- 541 of the
textbook. Ask the students to get the slopes and graph each example on the board. Guide the students to realize that the slopes can be used to determine whether the two lines are intersecting or parallel.
2. State and prove Theorem 7-1 and Theorem 7-3 with proper illustrations. State also the converse of the two
Theorems.
3. Working in pairs, ask the students to graph the following on their grid paper: y= 2x+1 and y= 1/2x +3.
Ask them to graph the two lines on one Cartesian plane. Discuss the graphs and lead them to the idea that the
slope of perpendicular lines are negative reciprocals of each other. State Theorem 7-5.
4. Discuss Examples 4 and 5 on pages 543-545.
5. Ask the students to consolidate the discussion. Then let them answer Mental Mathematics of Exercise 7.4 on
page 546.
C. Assignment
Prove Theorem 7-4 and Theorem 7-5
Session 2
A. Preliminary Activity
1. Ask somebody to explain the proofs of Theorem 7-4 and 7-5. Give the incentives to those who could prove
each theorem.
Page 161
2. Give the following reinforcement exercises.
Determine if the pair of lines are parallel, perpendicular or neither.
a. y= 6x+3 b. y= 3/4x- 5
y= 6x-1 3x-y = 6
c. 4x-y = 7 d. 2x+ y = 6
x+4y= 7 3x-y= 6
e. 2x+ 5y = 8 f. x-4y = 8
5x-2y= 7 2x- 4y = 8
Plot the following points and connect them from A to D.
A(1, 3) B (5, 1) C(7, 5) D(3, 7)
B. Follow-up Lessons
1. By using a protractor and a ruler, help the students see that the figure is actually square.
2. Ask them to get the slopes of pairs of opposite sides and also the slopes of pairs of consecutive sides. Let
them realize that the slopes can be used to determine whether a polygon is a square, rectangle, parallelogram,
or a right triangle.
3. Discuss the exercises in Written Mathematics of Exercise 7.4 on pages 546- 547 (numbers 1 and 2, 5 and 6)
of the textbook.
C. Assignment
Work on Written Mathematics of Exercise 7. 4 (numbers 3,4 and 7) on pages 546- 547 of the textbook.
Session 3
A. Preliminary Activities
1. Check the assignment
2. Make a brief recall of the point- slope form and standard form of a line.
Page 162
3. Give the following reinforcement exercises:
Determine if the quadrilateral with the following vertices is a parallelogram:
P (-4, 5) , Q (4, 6) , R (8,-1) , S (-2,-4)
Determine if the triangle with the given vertices is a right triangle.
R (-4,-5) , V (1, 0) , Q (-6,-3)
B. Follow up Lesson
1. Through the experience, present the solutions to written Mathematics of Exercise 7.4 (numbers 8-14) on page 547 of the textbook.
2. Ask the class to work in pairs in solving Written Mathematics of Exercise 7.4 (numbers 15-17) on page 547 of the textbook.
C. Checking for Understanding
1. Determine if the pair of lines are parallel, perpendicular, or neither.
a. y = -5x + 7 b. 2 x + 6y – 8 = 0
y = 
x – 3 3 x + 9 y + 3 = 0
x – 3 3 x + 9 y + 3 = 0 c. 3 x + 5 y = 10 d. 4 x – 7 y = 8
3 x – 5 y = 2 7 x + 4 y = 14
2. (2, 7), (- 2, - 1), (6, 5) are vertices of a right triangle. Is this a right triangle?
3. (-2, 1), (3, 3), (5, 0), (-1, -2) are vertices of a quadrilateral. Is this a parallelogram?
4. Write an equation in standard form of the line passing through the given point and parallel to the given line.
(4, 5); 5 x – 3 y = - 15
5. Write an equation in standard form of the line passing through the given point and parallel to the given line.
(- 1, - 2); 3 x – 2 y = 6
6. The line containing points (- 6, k) ang (4, 3) is perpendicular to the line containing points (13, 1) and (9, k + 1). Find K.
D. Assignment
Work on Written Mathematics of Exercise 7.4(numbers 18-21) on page 547 of the texbook.
Lesson 7.5
The Distance Formula
(Textbook pages 548-555, 1 session)
I. Mathematical Concept and Skills
A. If x₁ and x₂ are the coordinates of A and B, respectively, on a horizontal number line, then the distance between A and B is AB = lx₁ - x₂l.
B. If y₁ and y₂ are the coordinates of C and D, respectively, on a vertical number line, then the distance between C and D is CD = ly₂ - y₁l.
C. The distance between two points P₁(x₁, y₁) and P₂(x₂, y₂) is 
II. Objectives
A. Find the distance between two points on the same horizontal or vertical line.
B. Derive the distance formula using the Pythagorean Theorem
Use the distance formula to determine the distance between any two points on the Cartesian Plane.
III. Values Integration
A. Practice self-assessment or introspection
IV. Materials
Graphing board, colored chalk
Meter stick, ruler
Acetate or prepared board work for generalizations and exercise
V. Instructional Strategies
A. Expositions
B. Guided Discovery
C. Consolidation
VI. Procedure
Session 1
A. Preliminary Activities
1. Check the assignment.
2. Ask the permission from the immediate superior in conducting the activity in a basketball court, field or open space near the classroom.
a. Construct an arrow diagram or a number line on the floor.
b. Divide the class equally into two groups. Decide who will be the first one to play and the one to ask questions.
c. During the game, the first group to play must assign members on each coordinate. Representative from the opponent will ask question. The answer will be determined by the player assigned to the coordinate being asked. Coaching from the members will not be allowed. At the count of 10 after the question was asked, he answer must be given by the player assigned on the coordinate. The player must sit down. Otherwise the score will be given to the other group and it will be their turn.
3. Provide a quick review on integers and absolute value.
B. Lesson Proper
1. Discuss how to find the distance between 2 points on a number line through inductive approach. Provide as many examples as you can until the student can generalize how to find the distance between two points on a number line.
2. Present examples 1 in pages 549-550. If necessary give 3 pairs of coordinates, say (1, - 5) and (-1, -6) and allow a student to graph it and another student to compute the distance.
3. Let them answer Mental Mathematics of Exercise 7.5 on page 553 of the textbook.
4. Guide them to derive a formula for finding the distance between two points using the Pythagorean Theorem.
5. Give an Example where two points are not on horizontal line or vertical line such as the following pairs of two points found in Example 2 on pages 551-552. Draw it in a graphing board for emphasis.
6. Make use of the distance formula to explain the solutions made for Example 1.
7. Let them work in pairs in Answering Written Mathematics of Exerices 7.5(numbers 1-10 on page 554 of the textbook.
C. Assignment
Work on Written Mathematics of Exercise 7.5 (numbers 11 – 20) on page 554 of the textbook.
Session 2
A. Preliminary Activitires
1. Check the assignment
2. Relate a brief account of Pythagoreas?
Pythagoras (ca.
540 B.C.)
Very little is known about Pythagoras. He was a leader of a secret society or school founded in the 6th century B.C. This group, known as Pythagoreans, considered it impious for a member to claim any discovery for himself. Instead, each new idea was attributed to their founder, Pythagoras. This was more than school ; it was a philosophy and a way of life . Every evening , each member of the Pythagorean society had to reflect on three questions:
a. What good have I done?
b. Where have I failed?
c. What have I not done that should have done?
Ask your students to reflect on the same questions. Every student can share his answer to a partner.
3. Give these reinforcement exercises.
Find the distance between the following points.
a. (-2, 1), (-8, 1) c. (3, 7), (-11, 7)
b. (-2, 5), (0, 5) d. (14, 2), (14, -5)
B. Follow –up Lesson.
1. Present Example 3 on pages 552-553. Guide the students in analyzing and giving conclusions. Emphasize that a graph helps facilitate solving a problem.
2. Use exposition in solving this problem:
Show that triangle with the vertices A (3, -6), B (8, -2), and C (-1, 1)is a right triangle. Find the perimeter of ∆ABC.
The students should be able to prove that ∆ABC is a right triangle by showing that the square of the longest side is equal to the sum of the squares of the other sides. To determine the perimeter, let them add the lengths of the 3 sides.
3. Form groups of 4. Assign these exercises to each group
Group number Written Mathematics D, page 555
1&6 number 31
2&7 number 32
3&8 number 33
4&9 number 34
5&10 number 35
Let them write the solution on a manila paper and be ready to discuss its solutions.
C. Checking for understanding
Find the distance between each pair of points.
1. A(-4, 2), B (5, 4)
2. B(1, 2),C(4, -2)
3. C(4, 2), D(-2, 10)
4. D(-6, -2), E(-5, 4)
5. E(-5, 0), F(-9, 6)
6. F(-2, 4), G(7, -8)
7-8. The distance between points (1, 1) and (4, y) is 5. Find all possible values for y.
9-10. Find the perimeter of the triangle whose vertices are G (-7, 5), C (-7, -7), and F (2, -7)
For fast learners, ask them to solve the Challenge Problems on page 555 of the textbook.
D. Assignment
Work in Written mathematics of Exercise 7.5(numbers 21-50) on pages 554-555 of the textbook.
Lesson 7.6
The Midpoint Formula
(textbook pages 556-563, 2 sessions)
I. Mathematical Concepts and Skills
A. The midpoint of a horizontal segment lies between the endpoints. Its ordinate is the ordinate of the endpoints and its abscissa is the average of the abscissas of the endpoints.
B. The midpoint of a vertical segment lies between the endpoints. Its abscissa is the abscissa of the endpoints and its ordinate is the average of the ordinate s of the endpoints.
C. If P₁ (x₁, y₁) and P₂ (x₂, y₂) are any two points in a coordinate plane, then the midpoint of P₁ P₂ has coordinates
II. Objectives
A. Device, state, and apply the midpoint formula
B. Determine the midpoint of the segment joining 2 given points
III. Values Integration
Strengthen faith in God
IV. Materials
Graphing board, colored chalk, and straight edge
Visual aids in different illustration
That will help device the midpoint formula
Visual aids on different examples
Manila paper and crayons or pens
V. Instructional Strategies
A. Guided Discovery
B. Deductive Approach
C. Exposition
D. Discussion
E. Consolidation and Practice
VI. Procedure
Session 1
A. Preliminary Activities
1. Start with a prayer. After praying, ask what can be obtained from praying. Ask the class to relate stories related to the power of prayers.
2. Check the assignments
3. Motivation
a. Present the following situation
A Painter plans to create an abstract piece of art. He wants to stars at the center of his rectangle canvas. He plots a scale drawing on the coordinate axes and he assigns the vertices of the rectangles as follows: A (0, 0), B (12, 0), C (12, 8), and D (0, 8). How can he locate the center of this rectangular canvass?
b. Use paper folding in illustrating midpoint of diagonals and sides of quadrilateral, and the center of a circle.
B. Lesson Proper
1. Show illustrations of horizontal and vertical segments drawer on Cartesian plane. For each segment, guide them how to determine the abscissa and the ordinate of the midpoint. Use the examples given in the textbook on pages 556-557.
2. Emphasize the following about midpoint of a segment
a. Midpoint M of segment AB lies on AB and AM = MB
b. The midpoint divides a segment into two segment s of equal lengths.
3. With proper illustration, guide the students in the deprivation of the Midpoint formula. You may use the presentation in the textbook on pages 557-558.
4. Discuss Example 1 on page 559. Ask them to go back to the problem posed at the star t of the discussion. Let them find out if their given answers are correct.
5. Ask the class to consolidate the discussion, then let them answer Mental Mathematics of Exercise 7.6 on page 561 of the textbook.
6. Let them work in pairs solving Written Mathematics of Exercise 7.6(A, numbers 1-5,and B,1-5) on pages 562 of the textbook.
C. Assignment
Work on Written Mathematics of Exercise 7.6 (A, numbers 6-10 and B, 6-10) on page 562 of the textbook.
Session 2
A. Preliminary Activities
1. Check the assignment.
2. Give those reinforcement exercises.
Find the midpt. Of the segment joining the given pairs of points.
a. (3, -6), (3, 8)
b. (-4, -6), (-2, -9)
c. (4, 8), (2, -8)
d. (7, -9),(4, 8)
e. (-5, -4), (-11, -22)
f. (2 ½ , 3) (4 ½ , 2 ½ )
If M is the midpt of PQ, determine the coordinates of Q.
a. M (1,-2), P (3, -6) B. M (-1, 5), P (3, 8)
B. Follow up Lesson
1. Briefly recall the following:
A. medians of triangle
B. Kinds of parallelograms
C. Distance formula
D. Perimeter
2. Discuss Example 2 on pages 560-561.
3. Form eight groups of students. Let each group solve one of the problems of Written Mathematics of Exercises 7.6 (C, numbers 1-4)
On page 562. Then let them discuss the solutions to the class.
C. Checking for Understanding
1. Find the midpoint of the segment with the given endpoints.
a. (27, -36), (77, -102)
b. (-4.72, 1.04), (-5.20, -5.36)
c. (-1 ⅓, -2), (⅓, 2)
d. (2x + y, -x +3y), (3y, -x +y)
2. If M is the midpoint of PQ, determine the coordinates of Q.
a. M (0, 0), P (-3, -1)
b. M (3 1/2 , -7) P (3, -8)
c. M (1 ½, ½ ), P (5, -5)
3. Solve the following.
The vertices of a quadrilateral are (2, -10), (-4, -2), (8, -5) and (2, 3). Show that the diagonals of a quadrilateral bisect each other.
For fast learners, let them solve the Challenge Problems on page 263.
D. Assignments
Work in Written Mathematics of Exercise 7.6 (C, numbers 5 and 6) on page 563 of the textbook.
Lesson 7.7
Coordinate Proofs of Geometric Theorems.
(Textbook pages 564-569, 3 sessions)
I. Mathematical Concepts and Skills
A. “Coordinate Geometry” is about proving theorems by means of placing a given figure in a proper position on the coordinate axes and using algebraic principles and procedures in the process.
B. The following are suggested to appropriately place a geometric figure in a coordinate plane:
1. Place one of the vertices of the geometric figure at the origin
2. Place the sides of the geometric figure along the coordinate axis.
II. Objectives
A. Place triangles and quadrilaterals on the Cartesian plane and find the missing coordinates.
B. Use coordinates proofs in verifying properties of triangles and quadrilaterals
III. Values Integration
Show skills in finding ways and means in doing things
IV. Materials
Graphing paper, colored chalk, straightedge
Grid papers
Visual aids on most common ways of placing geometric figures on the coordinate axes
Some properties of triangles and quadrilaterals
Manila paper and crayons or pentel pens
V. Instructional Strategies
a. Expositions
b. Discussions
c. Guided Proving
d. Consolidation and Practice
VI. Procedure
Session 1
A. Preliminary Activities
1. Administer a game on identifying the kinds of quadrilaterals and identifying its properties by plotting points,
a. Ask the students to form five groups. Give different sets of coordinates forming geometric figures, tell the students to plot on the Cartesian plane and identify the figure formed. The first group to identify the figure should state a property of the figure to earn a point. The group that can earn more points will be given incentives.
b. Tell them that the above activity is what we called the Coordinate Geometry. What do you mean by Coordinate Geometry? Entertain the student’s responses.
B. Lesson Proper
1. State that “Coordinate Geometry refers to the association of real numbers with points and this association makes it possible to prove theorems concerning plane figures by algebraic means. State the steps in proving geometric Theorems using coordinates.
2. Illustrate the most common way of placing geometric figures on the coordinate axes as show
a. For Isosceles triangle
b. For Quadrilaterals
3. Illustrate examples on finding the missing of figures without using new variables.
4. Ask the class to consolidate the discussion, then let them answer the mental Mathematics of Exercises 7.7 on page 567 of the textbook.
C. Assignment
Work on Written Mathematics of Exercise 7.7 (A, numbers 3 and 4) on page 568 of the textbook.
Session 2
A. Preliminary Activities
1. Check the assignment.
2. As a review, give the following exercises.
Solve for the distance between the following pairs of points.
a. (-2, -3), (6, 1) b. (3, 4), (-2, 1)
Find the midpoint of the segment with the given endpoints.
a. (4, 8) b. (-4, -7), (-9, 8)
3. Recall basic concepts related to quadrilateral and triangles, the slopes of paralle; and perpendicular lines.
B. Follow-up lesson
1. Illustrate the use of coordinate geometry in proving some properties of triangles. Use example 1 on pages 564 -566. Present other examples.
2. Give example of proving some properties of quadrilaterals using coordinate Geometry. Use Example 2 on pages 566- 567.
3. Let the class work in pairs in working on written Mathematics of Exercise 7.7 (b, #568-569) of the textbook
C. Checking for Understanding
Show a coordinate proof for the statement:
“The midpoint of the hypotenuse of any right triangle is equidistant from each of the 3 vertices
D. Assignment
Work in Written Mathematics of Exercises 7.7(B, #6-10) on page 569 of the textbook.
Lesson 7.8
Circles in the Coordinate Plane
(textbook pages 570-580, 3 sessions)
I. Mathematical Concepts and Skills
a. A circle is a set of coplanar points all of which are equidistant from a fixed point called center.
b. The standard equation of the circle with radius(r) and center at (0, 0) is
r³ = x² + y².
c. The standard equation of a circle with radius (r) and center at my point (h, k) is
r² = (x – h) + (y – k)
d. The general form of the equation of a circle is x² + y² + Dx + Ey = 0
III. Objectives
A. Recall the definition of a circle
B. State and derive the standard form of the equation of a circle with radius (r) and center (h, k)
C. Find the equation of a circle, given its center and radius and vice versa
D. Transform the equation of a circle in standard form to an equation in general form and vice versa
III. Values Integration
Discuss how to be flexible in life
IV. Materials
Compass
Meter stick
Acetate or prepared board work and illustration for generalizations and exercises
V. Instructional Strategies
a. Exposition
b. Guided Discovery
c. Consolidation and Practice
d. Cooperative work
VI. Procedure
Session 1
A. Preliminary Activities
1. Check the assignment.
2. Motivation
Ask the students if they know the rhyme: “Small Circle, Big Circle!”
Draw on the board while reciting to let them see the figure. Then Together, let the students draw on scratch paper.
“Small circle, Big circle!”
Small circle, small circle, Big circle
Small circle, small circle, Big circle
This is papa, this is Mama
Say Goodbye,
Six times Six, Six times Six, equals thirty six
Six times Six, Six times Six, equals say good bye
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