Math

Summer work for grade 7

INTERNATIONAL

SCHOOL OF ARTS & SCIENCES

GRADE 8 MATH SUMMER WORK A very useful Math Summer Sheet including Algebra and Geometry exercises that will recap all the concepts of Grade 8 and prepare well for Grade 9. In addition a number of challenging exercises are included to lead for more critical thinking

ISAS – Math department – middle school Algebra - Geometry

Math Summer Work

Algebra

Exersice 1: Develop the following binomials:

Examples- RECALL:

Solve:

1) (x+5)2

2) (x+2)3

3) (2x-5)2

4) (2x-3)3

5) (3x+2)2

6) (x2 − 12

y)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)

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

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

14) (x2+1)2

Exercise 2: Write each expression as a product of two binomials using the identity.

1) 1-9x2

2) 2 1x �

3) 24 16x �

4) 2 4x �

5) 2 9x �

6) 2 49x �

7) 2 100x �

8) 2 121x �

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: Factorize 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

Factor the following trinomials Exercise 8:

Exercise 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.

Write an algebraic statement for each line. Exercise 10:

Exercise 11: 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.

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.

1) 2xy + ab - yzb given expression

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

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

4) 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?

5) 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?

6) 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 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

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

Solve the variable in each equation. (Multiply out the brackets first if you have any):

Exercise 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

Extension Problems:

a) The area of this rectangle is 10cm2, 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 time its width and its perimeter is 24cm, what is its area?

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 – 65 x < 34 8) 8(11-2x) < 24 9) 10 > 8 -

32 x

10) 5(3x + 1) < -70 11) 15 - 85 x > 10 12) 24x – 32 < 8(5x – 12)

Simple Inequalities:

1) 12x – 17 > 19 2) 41 – 43 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 - 85 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.

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:

x 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:

3. Write down a polynomial in y of degree 4 with fractional coefficients.

Ans:

Perform each indicated operation:

1) 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 ���

8. 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).

Part 4: Subtract these polynomials.

18.) (6x + 14) 19.) (14x2 + 13x + 12)

- (9x + 5) - (5x2 + 12x + 7)

20.) (19x2 + 9x + 16)

- (7x2 + 20x + 4

21.) (17x2 + 7x - 14) - (-6x2 - 5x - 18)

22.) (-18x2 + 4x - 16) - (15x2 + 4x - 13)

Part 5: Multiplying Monomials

23.) )4(2 2xx 24.) )2(17 52 xx

25.) )4(3 23 xx� 26.) )2(12 2 xx ��

Part 6: Use the distributive property to find the product (multiply).

27.) )2(4 �x 28.) )12(3 2 �� x

29.) )72(6 2 �� xx 30.) )1(4 xx �

30.) )5(2 �� xx 31.) 3x²(4x³ – 5x + 10)

32.) � �1223 2 ��� xxx

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

Solve each equation.

1. 3 18x�

3. 5 8t �

5. 2 1 5x �

2. 5 35y

4. 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 � � d

12. 2 3 5t � d

14. 1 2 1 3 12w� � t

16. 3 2 5 9z � � !

Write each compound inequality as an absolute value inequality.

17. 7.3 7.3a� d d 18. 11 19md d

19. 28.6 29.2Fd d 20. 0.0015 0.0018td d

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

21. 22.

LAWS OF EXPONENTS:

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:

10) 12 3510 10x 11) 7 12a ax 12) 3 8c cx

13) 7 9d dx 14) 2 8e ex xx 15) 103 1030w wx

16) 6 5a bx 17) 10 10a bx 18) 12 19 11g g gx x

SIMPLIFY EACH PRODUCT:

19) � �� �2 3 22 4x x y 20) � �� �2 43 6a b ab c� 21) � �� �5 3 57 12q q r

22) � �� �8 411 10c c d� 23) � �� �10 2 5 39x z x y� 24)

� �� �6 2 58 7f g f g h� �

25) � �� �6 11 5 2 31.3 0.5a b c a bc 26) � �� �x y z r s ta b c a b c

SIMPLIFY EACH EXPRESSION:

27) � �32x 28) � �57a 29) � �413y 30) � � 1521w��

31) � �325 32) � �8723 33) � �45y� 34) � �234y

35) � �258c 36) � �393h� 37) � �84 6y d

38) � �45 6c h� 39) � �39 715h k� 40) � � � �5 29 3k k

41) � � � �26 5 23y x y z 42) � � � �2 33 34 2h g h� 43) � � � �2 74 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) � �2yz 49) � �3yx 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

1010

55) 17

14

44

56) 210

207

99

57) 12

2

y

y

�

58) 4

1

88

r

r

�

�

SIMPLIFY EACH EXPRESSION:

62) 6

xy

§ · ¨ ¸

© ¹ 63)

2

2

5cd

§ · ¨ ¸© ¹

64) 33

5

4dc

§ · ¨ ¸

© ¹

65) 4

6

3wg

§ · ¨ ¸

© ¹ 66)

36

3 5

4st r

§ ·� ¨ ¸

© ¹ 67)

211 6

18

2d fc

§ ·� ¨ ¸

© ¹

68) 342

4de

§ · ¨ ¸

© ¹ 69)

362rr 70)

6

3

4020ss

�

71) 18 5

11 3

217d ed e

72) 7 216

4w rwr

�

� 73)

5 5 5

2 3 4

a b ca b c

�

74) 4 14

9 5

4.20.6x yx y

75) 56

3

248tt

§ ·� ¨ ¸

© ¹ 76)

311 16

6 6

d fd f

§ · ¨ ¸

© ¹

77) 52

4

714dd

§ · ¨ ¸

© ¹

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

78) 3xx 79)

4yy 80)

3

3

x yxy

81) 4 2

2

z x yzxy

82) � �2yzz

83) � �23

3

39

y zxx

84) 1x

x

zz

�

85) 3

x x

y

zz

�

� 86) 3

xzy

§ · ¨ ¸

© ¹

-Try this fun online game!

http://www.softschools.com/math/games/algebra_practice.jsp

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_hotmath.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

ts5.xmlhttp://www.algebralab.org/practice/practice.aspx?file=Algebra_Exponen

inequalities

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

CONGRUENT TRIANGLES:

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

http://www.superteachertools.com/jeopardyx/jeopardy-review-game.php?gamefile=1322685871

TYPES OF TRIANGLES

http://www.factmonster.com/math/knowledgebox/player.html?movie=sfw41507

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

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

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

24

Geometry: - Remember some rules! -

25

26

Triangle Classification By the ANGLES of a Triangle

Equiangular Acute Right Obtuse

By the SIDES of a Triangle

Equilateral Isosceles Scalene

Exercise 1: Classify the triangle by is sides and angles. You must always be as specific as possible. A) B) C) Exercise 2: Explain why the triangle is a scalene right triangle.

_______________________________________________

_______________________________________________ Exercise 3: Find m�1 Exercise 4: Find m�Z EX 5: Find the value of x, y, and z.

X

56

Y Z

27

Exercise 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. What is the type of each angle?

Exercise 7: The variable expressions represent the angle measure of a triangle. Find the measure of each angle. Then classify the triangle by its angles. � �q� � 116xAm

� �q� � 23xBm � �q� � 15xCm

Objectives: 1) Use properties of isosceles and equilateral triangles. 2) Use properties of right triangles.

Exercise 8: Find the value of x and y and z. A) B) C) D) E) F)

:of each polygon Find the perimeter

6z . 24 .

3z + 2 8z - 33

28

29

x Objectives: Corresponding Parts and Congruent triangles Say whether the shapes are congruent or not.

30

Exercise 9: Given that 'MKL # 'JET, complete each statement. A) �L # ___________ B) MK # _______________ C) m�E = _________ D) ML = _______________ E) 'ETJ # _________ F) �JTE # _____________ Exercise 10: Find the value of x. A)

B)

Exercise 11: Identify any figures that can be proved congruent. Explain your reasoning. Write a congruence statement. A) B)

C) D)

31

Exercise12: Given: O is the midpoint of MQ and O is the midpoint of NP. Prove: Exercise 13:

Exercise 14: Is it possible to prove that the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Statements Reason 1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

1.____________________________

2.____________________________

3.____________________________

4.____________________________

5.____________________________

6___________________________________

32

A. B. C. D. E. Exercise 15: Given: AD || EC, BD # BC Prove: ∆ ABD # ∆ EBC

Statements Reasons 1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

33

x Use the labeled triangles to state the Pythagorean Theorem.

1) 2)

3) 4) 5)

25m 8cm 6) x 7) 8) 24 m x 10cm x 20cm Extra Exercises:

1. Which product equals 6x2 − 7x + 2 ? a. (6x + 1)(x − 7) b. (−6x + 1)(−x + 7) c. (2x − 1)(3x + 2) d. (−2x + 1)(−3x + 2)

t u

s

p

m

r

9

6

x z

25

7

8

17

m

34

2. The width of a rectangle is 13 feet less than twice its length. Which of the following shows an expression for the area of the rectangle?

a. 2l2 − 13l b. 2l2 − 13 c. 2l − 13l d. 6l − 26

3. The product of )32)(32( �� xx is:

a. 962 2 �� xx b. 94 2 �x c. 92 2 �x d. 19124 2 �� xx

4. The product of : )24)(24( nn �� is:

a. )216( 2n� b. 2416 n� c. n48� d. 228 n�

5. The side of a square is )52( �x what is its area: a. 2524 2 �� xx b. 2524 2 �� xx c. 2524 2 �� xx d. xx 20254 2 ��

1. The deer population of the Kaibab Plateau in Arizona from 1905 to 1930 can be estimated by the polynomial 3.13x4−0.13x5 +4000, where x is the number of years after 1900.

a. Find the degree of the polynomial b. Find the leading coefficient and the constant of the polynomial.

35

2. Simplify by finding the perimeter:

3. You are enlarging a photo that is 7 inches long and 5 inches wide. The length and width of the enlargement are x times the length and width of the original photo. The enlargement will have a 2 inch mat. Write a polynomial expression for the combined area of the enlargement and mat.

36

a. Write an expression for the area of the figure:

b. Write an expression for the area of the blue region.

37

Pythagorean Theorem + Distance and midpoints:

1. What is the length of the diagonal of a TV screen whose dimensions are 100 x 60 cm?

2. Given an isosceles right triangle with side 8m, what is the length of the hypotenuse?

3. Determine whether each set of measures can be the measures of the sides of a right triangle. Then state whether they form a Pythagorean triple.

(6,8,10) , (26,12,16)

4. Find OD:

5. Given the points A (−2, 6) and B (5, − 4) a. Graph the points A and B in the coordinate plan. b. Find the distance between AB c. Find the coordinates plan of M the midpoint of AB

38

.

6. Find the perimeter of triangle ABC to the nearest tenth if the coordinates are A(1, 4), B (-2, -1), and C(-3, -2).

7. If M(−2,1) is the midpoint of the line segment joining points A and B, and B has coordinates (−5, −3)

a. Find the coordinates of A. b. Find the length of AB.

8. Find The distance between AB and find the coordinate of M the midpoint of AB in each case:

a. A(-5,8) , B(-2,-6) b. A(-4,-6), B(-10, 5) c. A(-5, 6), B(-3,-4)

39