math booklet 7th

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Instituto Salesiano San Miguel 7th Grade Section: _____

Student: ___________________________________________________________

MATH BOOKLET 4TH

PARTIALThis booklet is a summary of the topics that will be covered in the 4th partial for MathematicsClass. Read the theory and develop all the activities proposed. Every work from this booklet shouldbe submitted in separate paper sheets.

Teacher: José Luis Ávila Betancourt


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Math Booklet 4th Partial

1

Review: September 7th to September 24th

Linear Equations

Set of Exercises #1

Activity 1

Solve these equations (find the value of x):

a) 5x + 1 = 31 b) 3x – 1 = 8 c) 7x = 60 + 2x

d) 3x = 72 – 3x e) 6x + 4 = 20 – 2x f) 6x + 3 = 23 + x

Summary: To solve equations, use the addition/multiplication principles.

1. Parentheses by using the distributive property. If no fractions, combine like terms.

2. Denominators: Multiply each side of equation by common denominator.

Decimals: Multiply each side of equation by 10, 100, 1000, etc. COMBINE LIKE TERMS.

BEFORE NEXT STEP EACH SIDE SHOULD BE NO MORE COMPLICATED THAN:

“4x – 8”

3. Signs (addition or subtraction) by using the addition principle (add opposites).

Get variable terms on one side of the equation and all constant terms on the

other side. Goal: Each side of equation is no more complicated than

“4x = -9.”

4. Coefficients by dividing by coefficient (BY SAME NUMBER). Goal: x = number

We can summarize the process of adding opposites or multiplying by the multiplicative

inverse by using transposing. In this case, instead of writing the same operation on both

sides of the equation, we transpose the number to the other side of the equation with the

inverse operation or sign.

lgebra


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g) 5x + 4 = 2x + 17 h) 5x + 11 = 20x – 64 i) 28 – x = 17 + 3x

j) 6x + 7 = 8x – 13 k) 13.7b – 6.5 = -2.3b + 8.3 l) 28 – 2.2y = 11.6y + 262.6

Activity 2

Solve these equations with brackets (multiply out the brackets first):

a) 5(x + 2) = 25 b) 2(2x + 10) = 40

c) 3(2x – 5) = 21 d) 4(5x –3) = 7(2x + 3)

e) 3(4 + x) = 5(10 + x) f) 2(3x – 4) = 4x + 3

g) 4(2a – 8) =1

7(49a + 70) h) -8(4 + 9x) = 7(-2 – 11x)

i) 3(4 - 2x) - (12 + 3x) = 18 j) 4[7x - 3(2x + 7)] = 8(-4 - (0.5)x)

Activity 3

Solve these equations with brackets (simplify common terms at first and multiply the

expressions times the LCD):

a)3n 2

5 7

10 b)

3

2 y y 4

1

2 y

c)3

13

4

1 ww d)

4

3

6

5

x

e) x x x2

13

4

1 f) 1

4

1

3

2 mm

g)2

32

3

1 mm h)

4

3

6

5

x

i)

3

2

4

3t j)

2

1

4

3 x

k) 2124

3 x l) 513

3

2 x

m) t t t 5

11

5

2

2

1 n)

5

22

3

2

5

1 mm


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3

Step-by-step application of linear equations to solve practical word problems:

a)

The sum of two numbers is 25. One of the numbers exceeds the other by 9. Find the

numbers.

Solution:

Then the other number = x + 9

Let the number be x.

Sum of two numbers = 25

According to question, x + x + 9 = 25

⇒ 2x + 9 = 25⇒ 2x = 25 - 9 (transposing 9 to the R.H.S changes to -9)⇒ 2x = 16⇒ 2x/2 = 16/2 (divide by 2 on both the sides)⇒ x = 8Therefore, x + 9 = 8 + 9 = 17

Therefore, the two numbers are 8 and 17.

b) The difference between the two numbers is 48. The ratio of the two numbers is 7:3.

What are the two numbers?

Solution:

Let the common ratio be x.

Let the common ratio be x.

Their difference = 48

According to the question,

7x - 3x = 48

Steps involved in solving a linear equation word problem:

Read the problem carefully and note what is given and what is required and what is

given.

Denote the unknown value (s) by the variables as x, y, …….

Translate the problem to algebraic language or mathematical statements.

Form the linear equation in one variable using the conditions given in the problems.

Solve the equation for the unknown.

Verify to be sure whether the answer satisfies the conditions of the problem.


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⇒ 4x = 48⇒ x = 48/4⇒ x = 12Therefore, 7x = 7 × 12 = 84

3x = 3 × 12 = 36

Therefore, the two numbers are 84 and 36.

c)

The length of a rectangle is twice its breadth. If the perimeter is 72 meters, find the

length and width of the rectangle.

Solution:

Let the width of the rectangle be x,

Then the length of the rectangle = 2x

Perimeter of the rectangle = 72Therefore, according to the question, 2(x + 2x) = 72

⇒ 2 × 3x = 72⇒ 6x = 72⇒ x = 72/6⇒ x = 12

We know, length of the rectangle = 2x

= 2 × 12 = 24

Therefore, length of the rectangle is 24 m and breadth of the rectangle is 12 m.

d) Aaron is 5 years younger than Ron. Four years later, Ron will be twice as old as

Aaron. Find their present ages.

Solution:

Let Ron’s present age be x.

Then Aaron’s present age = x - 5

After 4 years Ron’s age = x + 4, Aaron’s age x - 5 + 4.

According to the question;Ron will be twice as old as Aaron.

Therefore, x + 4 = 2(x - 5 + 4)

⇒ x + 4 = 2(x - 1)⇒ x + 4 = 2x - 2⇒ x + 4 = 2x - 2⇒ x - 2x = -2 - 4


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⇒ -x = -6⇒ x = 6

Therefore, Aaron’s present age = x - 5 = 6 - 5 = 1

Therefore, present age of Ron = 6 years and present age of Aaron = 1 year.

Activity 4

Extension Problems

a)

The area of this rectangle is 10 cm2, find the value of x and use it to find the length

and the width of the rectangle.

4x + 2

10x – 1

b)

If the length of a rectangle is three times its width and its perimeter is 24cm, what

is its area?

c) Three times the greatest of three consecutive even integers exceeds twice the least

by 38. Find the integers.

d)

The difference of two numbers is 12. Two fifths of the greater number is six more

than one third of the lesser number. Find both numbers.

e) Robert’s father is 4 times as old as Robert. After 5 years, father will be three times

as old as Robert. Find their present ages.

f) The sum of two consecutive multiples of 5 is 55. Find these multiples.

g) My mother is 12 years more than twice my age. After 8 years, my mother’s age will

be 20 years less than three times my age. Find my age and my mother’s age.

h) Adman’s father is 49 years old. He is 5 years older than four times Adman’s ag e.

What is Adman’s age?


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September 28th to October 2nd

Set of PointsPoints, Lines and Planes

Picture (1) shows a hand with three fingers stretched in

different directions.

Picture (2) represents the three directions by using arrows.

The part with the arrows start is called point .

A point represents a location and it doesn’t have

direction, neither extension.

If a point is moved in the same direction it will draw a

straight line.

If the point moves backward and forward infinitely it

would complete a straight line. A straight line doesn’t

have any width, it extends infinitely in two directions, and

it doesn’t have beginning neither end. (Picture 3)

If there is any change of direction in the pathway of the

point, generally we draw a non-straight line. (Picture 4)

From here, the word line will be used instead of

straight line.

If a line moves in another direction it draws a plane. The

displacement of the line in order to form a plane is the

same to the displacement of the point to form the line. A

plane has two directions but it doesn’t have any

thickness. (Picture 5)

When a non-straight line is moved it usually draws a

surface. (Picture 6)

Picture (1)

Picture (2)

Picture (3)

Picture (4)

Picture (5)

Picture (6)

Geometry


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Lines can be named and labeled two different ways:

1. We use a lowercase, cursive letter of the alphabet to name and label

a line. We usually use the letters l, m, and n; however, anylowercase, script letter of the alphabet may be used. The graphic to

the right shows line m.

2.

We can identify any two points on the line to name and label a line.

We name the line by the two points. We use either the word “line”

or the double-headed arrow above the two named points to signify

the line. Note that any two points identify a unique line. Thus, the

graphic to the right shows line PQ, line QP, ⃡ or ⃡ .

Points

A point indicates position it has no dimensions. To draw a point, we

generally draw a small, filled-in circle such as the graphic to the right. Weuse capital letters of the alphabet to name and label a point. For example,

the point to the right is Point P. We usually use the letters P, R, Q, or A, B,

C, and X, Y, Z to name points; however, any capital letter of the alphabet

may be used.

Lines

A line only possesses length. Lines go forever and ever in both directions

but have no height. To draw a line, we draw a line that is very thin (to

represent no thickness) and we place arrows on each end of the line to

signify that lines go forever and ever in both directions (see the graphic to

the right). Another definition of a line is “an infinite collection of collinear

points,” but this definition is somewhat circular.

m

P

Q

Planes

A plane is a flat surface consisting of infinitely many points. To draw a

plane, we generally draw a parallelogram such as the graphic to the right.

A sheet of paper is a good representation of a plane, as long as we

remember that a plane has no width and goes forever and ever in all

directions. We use lowercase, letters of the Greek alphabet to name and

label a plane. For example, the plane to the right is plane alpha (α).

α


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Points on the same line are said to be collinear!

Points that lie on the same plane are said to be coplanar!

Another way to name a plane is by giving _____ points on the plane!

Example 1

Name Lines and Planes.

Example 2

Model Points, Lines and Planes.


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Set of Exercises #2

Activity 1

a.

Which are the possible names for the next line.

____________ ____________ ____________ ____________ ____________ ____________ ____________

____________ ____________ ____________ ____________ ____________ ____________

b. Write whether the show part on the next pictures is a point, a line, or a plane.

c.

State whether each is best modelled by a point, line, or plane.

(1) a knot in a piece of thread

(2)

a piece of cloth

(3)

the corner of a room

(4)

the telecommunications beam to a satellite in space

(5)

the crease in a folded sheet of wrapping paper

(6)

an ice skating rink

d. List three real world objects that could be modeled using a point, a line, and a

plane.

(1)

Point: ___________________________________________________________________________

(2)

Line: ____________________________________________________________________________

(3)

Plane: ___________________________________________________________________________

e. Draw the next lines.

(1) ⃡ (2) ⃡ (3) ⃡ (4) ⃡ (5) ⃡


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Activity 2

Refer to the next diagram.

a)

Name line ⃡ in another way. _______________

b)

Give two other names for plane Q.

__________________________________________________

c)

Why is EBD not an acceptable name for

plane Q?

__________________________________________________

__________________________________________________

d)

Are the following sets of points collinear?

(1)

E , B, and F

(2) Line ⃡ and line ⃡

(3)

Line ⃡ and line ⃡

(4) Line ⃡ and line ⃡

(5) F , A, B, and C

(6)

F , A, B, and D (7) Plane Q and line ⃡

(8)

Line ⃡ and line ⃡

Activity 3

a)

Name a point that is collinear with the given points.

B and E: _________

C and H: _________

D and G: _________

A and C: _________

H and E: _________

G and B: _________


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Activity 4

a)

Name a point that is coplanar with the given points.

M, N, R: ______

M, N, O: ______

M, T, Q: ______

Q, T, R: ______

T, R, S: ______

Q, S, O: ______

Activity 5

a)

Find the intersection of the following lines and planes in the figure below.

(1)

Line ⃡ and line ⃡ ____________

(2) Planes GLM and LPN ____________

(3)

Planes GHPN and KJP ___________

(4)

Planes HJN and GKL ___________

(5) Line ⃡ and plane KJN __________

(6)

Line ⃡ and plane GHL _________

Activity 6

a)

Refer to the next diagram:

(1)

Name plane P in another way

(2)

What is the intersection of planes P and Q?

(3) Are A, B, and C collinear?

(4)

Are D, A, B, and C coplanar?

(5)

What is the intersection of line ⃡ and line ⃡

(6) Are planes P and Q coplanar

(7)

Are line ⃡ and plane Q coplanar?

(8) Are B and C collinear?


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October 12th to October 16th

Set of PointsRays and Line Segments

We name a ray by the one endpoint and any other point on the ray. We place a ray

symbol (a one-ended arrow ) above the two points. Be careful when writing the ray.

In line ⃡ above, we could have written line ⃡ , and we would be referring to the

same line. However, ray and ray are entirely different rays. It is best to use the

ray symbol above a ray’s name so it is clear which of the two points the ray’s endpoint

is. Thus, the graphic above shows ray PQ or .

The graphic to the right shows the line segment PQ or .

Unlike a line, a line segment , or segment , can be measured because it has two

endpoints. A segment with endpoints A and B can be named as or . The

length or measure of is written as AB. The length of a segment is only asprecise as the smallest unit on the measuring device.

Geometry

RAY

A ray is a portion of a line with a distinct start but no end. Some people

define a ray as half of a line but this is not entirely correct because a line

goes on forever and ever (to infinity) in both directions. There is no such

number as “half of infinity.”

LINE SEGMENT

A line segment is a portion of a line with a distinct start and a distinct

end. Thus, a line segment has two endpoints. We name a line segment by

listing the two endpoints. We use either the word segment or a segment

line above the two endpoints.


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Measures are real numbers, so all arithmetic operations can be used with

them. You know that the whole usually equals the sum of its parts. That is

also true of line segments in geometry.

Recall that for any two real numbers and , there isa real number between a and b such that .This relationship also applies to points on a line and

is called betweenness of points. Point M is between

points P and Q if and only if P , Q, and M are collinear

and .

Example 1

Length in Metric Units.

Example 2

Length in Customary Units.


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Look at the figure in part a of example 4. Note that and have the samemeasure. When segments have the same measure, they are said to be

congruents.

Example 3

Find Measurements.


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Constructions are methods of creating geometric figures without the

benefit of measuring tools. Generally, only a pencil, straightedge, and a

compass are used in constructions.

You can construct a segment that is congruent to a given segment by

using compass and straightedge.

Set of Exercises #3

Activity 1

a.

Draw line segments with the next lengths.

(1)

5 cm (2)

2 cm (3)

6.5 cm (4)

2.9 cm (5)

2.1 cm

b.

By using compass and straightedge, construct congruent segments to each one

of the line segments above.


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DISTANCE BETWEEN TWO POINTS The coordinates of the endpoints of a

segment can be used to find the length of the segment. Because the distance

from A to B is the same distance from B to A, the order in which you name

the endpoints makes no difference.

MIDPOINT OF A SEGMENT The midpoint of a segment is the point halfwaybetween the endpoints of the segment. If X is the midpoint of a segment ,then A, X , and B are collinear and .

The coordinate of the midpoint of a segment whose endpoints have

coordinates a and b is

.

Activity 2

Example 4

Find Distance in a Number Line.

Example 5

Find Coordinates of Midpoint.


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Any segment, line, or plane that intersects a segment

at its midpoint is called a segment bisector. In the

next figure, M is the midpoint of . Plane N , ,

, and point M are all bisectors of . We say that

they bisect .

You can construct a line that bisects a segment without measuring to find

the midpoint of the given segment.


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Set of Exercises #4

Activity 1

By using compass and straightedge (without measuring), construct what is asked

below.

(1)

Find the midpoint of the next line segments respectively.

(2)

Draw a line bisector to each one of the segments above.

(3) Find and represent a point so

.

(4)

Find and represent a point so

.

(5) Find and represent a point so

.

Activity 2

a.

Answer the next statements.

(1)

If AB = 7 and BC = 10, is it necessary that AC = 17 ? Explain your answer.

_______________________________________________________________________________________

(2)

If HJ = 20, JK = 15 and HK = 35, what do we know about A, B and C ?

_______________________________________________________________________________________

(3)

If A, B and C are collinear, AB = 13 y BC = 17 , does AC necessary equal 21?

_______________________________________________________________________________________

b. Find the length of the segment if B is between A and C . (Recommendation, draw

the line segments)

(1)

(2)

(3)


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Activity 3

Make a list including the collinear points in the next picture. Make another list with

sets of three non-collinear points.

Collinear points Sets of three coplanar and

non-collinear points.

Activity 4

Match with a segment the same name points.

Activity 5

Draw in the space, by using perspective, a representation of at least four different

planes that intersect all in the same line.


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October 19th to October 23rd

Angles

Geometry


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A ray that divides an angle into two congruent angles is

called an angle bisector. If is the angle bisector of ∡RPS ,then point Q lies in the interior of ∡ RPS and ∡ ∡.

You can construct the angle bisector of any angle without

knowing the measure of the angle.

Set of Exercises #5

Activity 1

a.

By using a protractor, draw the next angles.

(1) ∡45° (2) ∡27° (3) ∡180° (4) ∡106° (5) ∡95°

(6) ∡15° (7) ∡150° (8) ∡270° (9) ∡360° (10) ∡60°

b. By using compass and straightedge (without measuring), construct congruent

angles to each one of the angles above. (10 constructions)

c.

By using compass and straightedge (without measuring), construct the angle

bisector of each one of the angles in a. (10 constructions)

Activity 2

a. Construct by using a protractor 5 acute angles, 3 right angles, and 5 obtuse angles.


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October 26th to October 30th

PARALLEL LINES

Lines are parallel if they are always the same distance apart (called "equidistant"), and will never meet. Just remember: Always the same distance apa

and never touching . The red line is parallel to the blue line in both these cases:

Example 1 Example 2

Parallel lines also point in the same direction.
https://www.mathsisfun.com/geometry/line.htmlhttps://www.mathsisfun.com/geometry/line.html

math booklet 7th

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