Math Summer Work

Algebra

Exersice 1: Develop the following binomials:

1) (x+5)2

2) (x+2)3

3) (2x-5)2

4) (2x-3)3

5) (3x+2)2

6) (x2 − 1

2𝑦)2

7) (2x-5)3

8) (6x-7)(6x+7)

9) (1-4p)(1+4p)

10)(11v2-8) (11v2+8)

11)(x-13)(y+13)

Grade 8

Math Summer Work

Name: ___________________

International School of Arts and Sciences ISAS

2012 – 2013

12)(-y-3x)(-y+3x)

13)(7m2-3p) (7m2+3p)

14)(x2-x+1)2

Exercise 2: Put into product of factors/Factorize

1) 𝑥2 − 169

2) 𝑦3 − 125

3) 8x3-27

4) 1-9𝑥2

5) 343𝑚3 + 64𝑛3

6) 216y𝑥3 + 125𝑦4

Exercise 3: Find the coefficient of x in the expansion of (3x-7)2

Exercise 4:Explain how to find the product of 39 and 41 mentally.

(Hint: Write 39 as 40 - 1 and 41 as 40 + 1)

Exercise 5: Write as a perfect square or cube:

1)x3+9x2 + 27x+27

2) x2-4x+4

3)64x2+16x+1

4) x3-15x2 + 75x-125

Exercise 6: Expand then Reduce:

1) 𝑥 − 3 2+ 𝑥 + 3 2

2) 𝑦 − 2 2+ 𝑦 − 1 2-(𝑦 − 2)(𝑦 − 1)

3)4 𝑦 − 2 (𝑦 + 1)-2(y-2)(y+1)

4) 𝑥 − 4 3+(x-3)2

Exercise 7: Facorize using the GCF

1)34𝑥𝑦3 − 28𝑥3𝑦 − 14𝑥𝑦

2)280x2-140x

3)216𝑥2 − 56𝑦2

4)22x+11y-33z

5)3x(4y-1)+2z(4y-1)

6)(3x-7)2+(3x-7)4

Ex: 8 Word Problems

Remember to show all 5 parts.

1. Label the unknowns.

2. Set up the equation.

3. Solve the equation.

4. Find the value for the unknowns.

5. Check.

Remember: To find the value of a PERIMETER of a figure ,add the measures of the sides. To find the area of A square: side x side A rectangle: length x width

A triangle:1

2(base x height)

A parallelogram: base x height

Ex 9. Work all problems in the space provided

1. Find two consecutive integers whose sum is 45.

2. Find three consecutive even integers whose sum is 72.

3. Find two consecutive odd integers whose sum is –88.

4. Find four consecutive odd integers whose sum is 56.

5. Find two consecutive even integers such that the sum of the larger and twice the smaller is 62.

6. Find three consecutive integers such that the sum of twice the smallest and 3 times the largest is 126.

7. Seven times a number is equal to 12 more than 3 times the number. Find the

number.

8. Ten more than 6 times a number is 4 less than 4 times the number.

9. The second of two numbers is 4 times the first. Their sum is 50. Find the numbers.

10. The second of two numbers is 5 more than twice the first. Their sum is 80. Find the numbers.

11. The perimeter of a rectangle is 24 inches. Find the dimensions if its length

is 3 inches greater than its width.

12. Find the measures of the angles of a triangle if the angles are represented by x, 4x, and 4x.

13. The perimeter of a triangle is 51 centimeters. The lengths of its sides are consecutive odd integers. Find the lengths of all three sides.

14. Two-thirds times a number plus 7 equals 7 minus the number. Find the number.

Exercise 10: Write an algebraic expression for each real world example.

1. Concert tickets cost $15.50 each. If n tickets are bought, what is the total cost?

2. Movies are rented for $2.50 each. What is the charge to rent x number of movies?

3. A cellular phone company charges $.34 per minute flat rate.

4. A car contains g number of gallons of gasoline. Write an algebraic expression for the

amount that the tank holds after 5 gallons are added.

In questions 5 – 10, first identify what the variable represents, then write the expression.

5. A TV repairman charges $35 for a house call and $25 per hour to work on your TV.

6. A cellular phone company charges $16.95 basic rate plus $.35 per minute of airtime.

7. A utility company charges $12 for the first 100 kilowatt-hours of electricity used by a

household and $.14 for each kilowatt-hour over 100.

8. A plumber charges $20 service call plus $5 per hour to fix a sink.

9. The phone company charges $.15 per minute flat rate for long distance phone calls.

10. A company charges $19.95 for Internet service up to 60 minutes of log on time. After 60

minutes the company charges $.10 per minute for log on time.

Ex 11: EVALUATE ALGEBRAIC EXPRESSIONS

When you are asked to evaluate algebraic expressions, you will follow these steps:

1) substitute the given values for the variables

2) simplify the expression following the order of operations

Let x = 2 y = -1 z = 3 a = 0 b = -5.

a) 2xy + ab - yzb given expression

b) 3 - b(2x + 5b) given expression

c) 4xy5 - 2ab + (z + y)3 given expression

d) 9/5C + 32 = F is the formula to convert degrees in Celsius (C) to degrees in Fahrenheit (F). (The U.S. uses Fahrenheit while the rest of the world uses Celsius. Therefore, this formula is a wise one to know if you plan on traveling and want to know what the day's temperature is.)

The local news in Barcelona, Spain, announces it is 35 degrees Celsius. How will

you dress?

e) The formula for interest earned is I = p - p(1 + rt) where I is the interest, p is the principal, r is the interest rate and t is the time.

How much interest will you earn if you invest $2000 at 2% interest rate for 3

years?

f) The formula for work of an object is W = 0.5m(V2)2 - 0.5m(V1)2 where m is

mass, V2 is the second velocity and V1 is the first velocity.

How much work is done if m = 40, V2 = 30 and V1 = 20?

Aim

To solve linear equations with integer coefficients.

Exercise 12

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

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

Exercise 13: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

Exercise 14: Solve and Graph:

1) 3/5 x + 9 < 12 2) 4(2-3x) < 32 3) -21 < 7x – 144

4) 5x – 2 < 7x – 8 5) 11x – 5 > 15x + 3 6) 8x + 3 > 12x + 13

7) 24 – (5/6)x < 34 8) 8(11-2x) < 24 9) 10 > 8 - ⅔ x

10) 5(3x + 1) < -70 11) 15 - ⅝ x > 10 12) 24x – 32 < 8(5x – 12)

Ex15: Solve and graph the inequalities

1) 12x – 17 > 19 2) 41 – ¾ x < 53

3) 14x - 2 > 20x + 10 4) 8(5x – 4) – 6(3x + 5) < -7

5) 6(4x -2) > 5(7x + 2) 6) 16 < 5x – 4

7) 6(6x – 3) + 4(7 – 12x) > 28 8) -24 < 26 - ⅝ x

9) 8(7x + 5) > 5(4x + 8) 10) 4(7x +3) – (16x – 13) > 17

11) 7(6x – 4) < 4(3x – 7) 12) (15x – 8) – (19x + 8) < -14

13) A submarine is 150 feet below sea level. It has rock formations above and below it,

and should not change its depth by more than 27 feet. If d is its distance below sea level,

write an absolute value inequality that represents this situation and solve it.

Ex 16:

1. Which of the following is a polynomial and which is simply an algebraic expression?

a) 3x3-4x+12

b) 5x2-11w3 y0

c) 5𝑥𝑦 − 12𝑦−2

2. Identify each of the following expressions as monomial, binomial, trinomial,

or polynomial (more than three terms).

(a)

(b)

a2 + b2

x0 −2x+3y

Ans:

Ans:

(c)

ax2 − y2

Ans:

3. For the polynomial :

4𝑥3 − 12𝑥2 − 0.5𝑥 −1

(a) How many terms does it have? Ans:

(b) Write down the constant term. Ans:

(c) What is the degree of the polynomial? Ans:

(d) What is the coefficient of x2 ? Ans:

(e) Find its value when x = 2. Ans:

4. Write down a polynomial in y of degree 4 with

fractional coefficients.

Ans:

Perform each indicated operation:

5) Circle the problems that are in standard form. If it is not in

standard form, re-write in standard form.

a. 23 11xx

b. 32 3432 xxx

c. 24 2173 xxx

d. 2231 xx

6. Given: 12252 23 xxx

How many terms are there? ____

What is the coefficient of the 3rd term? ____

What is the constant? _____

Part 3: Add these polynomials. Only combine things that are

alike (have the same exponent).

7) 14x + 5 8) 10x + 12

+10x + 5 + 6x + 20

9) 1117 2 x

+ 118 2 x

10) ( )13107()121219 22 xxxx

11) ( )181519()764 22 xxxx

12) )51719()131520( 22 xxxx

13) )141510()49( 4556 xxxx

14) ( )13107()129 22 xxx

15) )6209()695( 2636 xxxxxx

Part 4: Subtract these polynomials.

16) (6x + 14)

- (9x + 5) 17) (14x2 + 13x + 12)

- (7x2 + 20x + 4

18) (17x2 + 7x - 14) - (-6x2 - 5x - 18)

19) (-18x2 + 4x - 16) - (15x2 + 4x - 13)

Part 5: Multiplying Monomials

20) )4(2 2xx 21) )2(17 52 xx

22) )4(3 23 xx 23) )2(12 2 xx

Part 6: Use the distributive property (rainbow) to find the

product (multiply).

24) )2(4 x 25) )12(3 2 x

26) )72(6 2 xx 27) )1(4 xx

28) )5(2 xx 29) 3x²(4x³ – 5x + 10)

30) 1223 2 xxx

Part 7: Use division and the distributive property to simplify.

Divide EVERY term.

31) 5

1015 x 32)

2

106 2 x

33) 3

2118 2

xx

34) 7

702814 23 xx 35)

2

24

5

1520

x

xx

36) x

xxx 73 34

Ex 17 For each of the following problems, indicate whether you will

use “and” or “or” and solve the problem.

1. | 2x – 5 | > 7 2. | x – 9 | < 4

3. | 2x – 1 | < -3 4. | x + 4 | - 6 < 9

5. | 2x – 1 | - 7 > -3 6. | 5x + 6 | + 4 < 1

7. | 3x – 4 | + 9 > 5

Ex 18: Solve each equation.

1. 3 18x

2. 5 8t

3. 2 1 5x 4. 5 35y

5. 3 7 12z

6. 4 2 5 9y

Solve each equation. Check for extraneous solutions.

7. 5 3 7x x 8. 2 3 3 2t t

9. 4 3 2 5w 10. 2 1 3 2z z

Solve each inequality. Graph the solution.

11. 5 3 15y

13. 4 3 9b

15. 2 4 1 5 1x

12. 2 3 5t

14. 1

2 1 3 12

w

16. 3 2 5 9z

Write each compound inequality as an absolute value inequality.

17. 7.3 7.3a 18. 11 19m

19. 28.6 29.2F 20. 0.0015 0.0018t

Write an absolute value equation or inequality to describe each graph.

21. 22.

LAWS OF EXPONENTS

Ex 19: Simplify. Write answers with positive exponents.

1) 15-4(158) 2) a7(a8)(a)

3) (3m4n6)(2mn)0(2m2n) 4) (-1x5y6)10

5) (5m3n)(-2mn3) 6) (7ab)(-a4b3)2(2a5b6)-1

7) (-9x3y4)(1/3x5)(-2y2)

SIMPLIFY EACH PRODUCT:

8) 12 3510 10 9)

7 12a a 10) 3 8c c

11) 7 9d d 12)

2 8e ex x 13) 103 1030w w

14) 6 5a b 15) 10 10a b 16)

12 19 11g g g

SIMPLIFY EACH PRODUCT:

17) 2 3 22 4x x y 18) 2 43 6a b ab c

19) 5 3 57 12q q r

20) 8 411 10c c d 21) 10 2 5 39x z x y 22) 6 2 58 7f g f g h

23) 6 11 5 2 31.3 0.5a b c a bc 24) 30) x y z r s ta b c a b c

SIMPLIFY EACH EXPRESSION:

25) 3

2x 26) (abc)0 27) 5

7a 28)(a2b3)4

29) 4

13y 30) 15

21w

31) 3

25 32) 8

723

33) 4

5y 34) 2

34y 35) 2

58c

36) 3

93h 37) 8

4 6y d 38) 4

5 6c h

39) 3

9 715h k 40) 5 2

9 3k k

41) 2

6 5 23y x y z 42) 2 3

3 34 2h g h 43) 2 7

4 6 6 314a b a c

Evaluate each monomial for x = 5, y = -1 , and z = 4

44) 4y 45) 33x 46) 22y 47) 2z

48) 2

yz 49) 3

yx 50) 2 2x z 51) xy

52) What is the area of a square with the length of a side equaling 53 ?a

53) What is the area of the rectangle with the width of 2 36 12 ?x and the length of x

SIMPLIFY EACH QUOTIENT AND THEN FIND THE VALUE OF THE RESULT:

54) 6

2

10

10 55)

17

14

4

4 56)

210

207

9

9 57)

12

2

y

y

58) 4

1

8

8

r

r

SIMPLIFY EACH EXPRESSION:

62)

6

x

y

63)

2

2

5c

d

64)

33

5

4d

c

65)

4

6

3w

g

66)

36

3 5

4s

t r

67)

211 6

18

2d f

c

68)

342

4

d

e

69)

36

2

r

r 70)

6

3

40

20

s

s

71) 18 5

11 3

21

7

d e

d e 72)

7 216

4

w r

wr

73)

5 5 5

2 3 4

a b c

a b c

74)

4 14

9 5

4.2

0.6

x y

x y 75)

56

3

24

8

t

t

76)

311 16

6 6

d f

d f

77)

52

4

7

14

d

d

EVALUATE EACH QUOTIENT IF X = 2 , Y = -2 , AND Z = 10 :

78) 3x

x 79)

4y

y 80)

3

3

x y

xy

81)

4 2

2

z x y

zxy 82)

2

yz

z 83)

23

3

3

9

y zx

x

84) 1x

x

z

z

85) 3

x x

y

z

z

86)

3

xz

y

Geometry:

Triangle Classification

By the ANGLES of a Triangle

Equiangular Acute Right Obtuse

By the SIDES of a Triangle

Equilateral Isosceles Scalene

EX 1: Classify the triangle by is sides and angles. You must always be as specific as

possible.

A) B) C)

EX 2: Explain why the triangle is a scalene right triangle.

_______________________________________________

_______________________________________________

EX 3: Find m1 EX 4: Find mZ EX 5: Find the value of

x, y, and z.

X

56

Y Z

Ex 6: The measure of one acute angle of a right triangle is five times the measure

of the other acute angles. Find the measure of each acute angle.

Ex 7: The variable expressions represent the angle measure of a triangle. Find the

measure of each angle.

Then classify the triangle by its angles.

116xAm

23xBm

15xCm

Objectives: 1) Use properties of isosceles and equilateral triangles.

2) Use properties of right triangles.

Ex 8: Find the value of x and y and z.

A) B)

C) D)

E) F)

6z .

24 .

3z + 2 8z - 33

Objectives: Corresponding Parts and Congruent triangles

Ex:9 Given that MKL JET, complete each statement.

A) L ___________ B) MK _______________

C) mE = _________ D) ML = _______________

E) ETJ _________ F) JTE _____________

Ex 10: Find the value of x.

A)

B)

EX 11: Identify any figures that can be proved congruent. Explain your reasoning.

Write a congruence statement.

A) B)

C) D)

Ex:12 Given: O is the midpoint of MQ

O is the midpoint of NP

Prove:

Ex 13:

Ex:14 Is it possible to prove that the triangles are congruent? If so, state the

postulate or theorem you would use. Explain your reasoning.

A. B.

C. D. E.

Statements Reason

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

B

A

C

F D

E

G

Ex 15:

Given: AD || EC

BD BC

Prove: ∆ ABD ∆ EBC

Construction 1：

(a) Draw a ABC.

(b) Construct the midpoint of segment BC.

(c) Label the midpoint as D.

(d) Draw a segment from vertex A to midpoint D.

(e) Construct the second median to side AC in the

same manner. Label the midpoint of segment AC as E.

(f) Label the point of intersection of these two medians as G.

(g) Construct the third median to side AB in the same manner. Label the midpoint of

segment AB as F.

Investigation 1：

1. When you construct the third median, does it pass through the point G?

____________________________________________________________________________

2. Move the vertices of the triangle. Does the way the medians intersect change?

____________________________________________________________________________

3. Write a conjecture about the way the medians intersect in a triangle.

____________________________________________________________________________

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

26

Construction 2:

(a) Measure the length of the segment AG. (b) Measure the length of the segment AD. (c) Measure the length of the segment GD. (d) Measure the lengths of other medians and their parts in the same

manner.

Investigation 2：

1. Calculate the ratiosAG

AD

BG

BE and

CG

CF, . What do you notice about the

ratios? Does this change when you move the triangle? ________________________________________________________________________

____

2. Calculate the ratiosAG

GD

BG

GE and

CG

GF, . Make a conjecture about the way

the centroid G that divides the medians of a triangle.

Algebra Links:

27

Polynomials http://www.quia.com/cb/42458.html?AP_rand=1252748981 http://www.quia.com/cm/25069.html?AP_rand=846450591 http://www.quia.com/pop/49672.html?AP_rand=1544097545 http://www.quia.com/jw/84714.html?AP_rand=1689863079 http://www.quia.com/rr/69387.html?AP_rand=1208533427 Difference of two squares

http://www.coolmath.com/crunchers/algebra-problems-factoring-

difference-two-squares.htm

http://www.mangahigh.com/en/maths_games/algebra/factorising/factorise

_xb_using_the_difference_of_two_squares

Perfect squares:

http://www.sporcle.com/games/TheCheese/perfect_squares

http://hotmath.com/hotmath_help/games/numbercop/numbercop_hotmat

h.swf

http://www.crctlessons.com/Perfect-Squares/perfect-squares-game.html

http://www.slidermath.com/rpoly/Prfectri.shtml

http://www.basic-mathematics.com/factoring-perfect-square-

trinomials.html

Algebraic expressions:

http://www.interactivestuff.org/sums4fun/substitute.html http://www.mathgoodies.com/worksheets/pdf/vol7_wks1.pdf http://www.aaaknow.com/g816b_x1.htm

http://www.edhelper.com/math/beginning_algebra2.htm Laws of exponents: http://www.slideshare.net/Josephil/laws-of-exponents-1677801

http://www.tpub.com/content/neets/14175/css/14175_173.htm http://oakroadsystems.com/math/expolaws.htm

http://www.mathsisfun.com/algebra/exponent-laws.html http://cnx.org/content/m18235/latest/ http://www.mathedpage.org/attc/lessons/ch.08/8.C-laws-of-

exponents.pdf http://www.algebralab.org/practice/practice.aspx?file=Algebra_Expon

ents5.xml

28

inequalities

http://www.bbc.co.uk/schools/ks3bitesize/maths/algebra/inequaliti

es_2.shtml

http://www.cool.math.com/school/subject2/lessons/S2U3L6GL.html

http://www.math-play.com/Inequality-Game.html http://education.jlab.org/sminequality/question.php?92447146 http://www.ltcconline.net/greenl/java/BasicAlgebra/inequalityGame/inequalities.html

Perimeter and Area :

http://www.wartgames.com/themes/math/areaandperimeter.html

http://www.mathplayground.com/area_perimeter.html

Geometry Links:

CONGRUENT TRIANGLES:

http://www.mangahigh.com/en/maths_games/shape/congruence/congrue

nt_triangles

http://www.superteachertools.com/jeopardyx/jeopardy-review-

game.php?gamefile=1322685871

TYPES OF TRIANGLES

http://www.factmonster.com/math/knowledgebox/player.html?movie=sfw4

1507

http://www.coolmath.com/reference/triangles-types.html

http://www.mangahigh.com/en/maths_games/shape/triangles/types_of_tri

angle

https://www.khanacademy.org/math/geometry/angles/v/angle-bisector-

theorem-examples

TAST:

http://quizlet.com/15332924/geometry-lesson-34-triangle-angle-sum-

theorem-flash-cards/

http://www.mathwarehouse.com/geometry/triangles/

http://www.math-play.com/Angles-Jeopardy/Angles-Jeopardy.html