Topic: Integers. Addition Subtraction Multiplication Division Same signs: Add & keep sign = = - 10.

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1 Topic: Integers Examples: Addition Subtraction Multiplication Division Same signs: Add & keep sign = = - 0 Different signs: Subtract & take sign of larger value = = - 5 Keep Change-Opposite = = = = = - 2 Same signs: Positive product ( + 7) ( + 8) = + 56 ( - 2) ( - 6) = + 2 Different signs: Negative product ( + ) ( - 9) = - 27 ( - 5) ( + 4) = - 20 Same signs: Positive quotient + 42 / + 6 = / - 8 = + Different signs: Negative quotient + 56 / - 7 = / + 2 = - 25 Recall the order of operations: Parentheses (or grouping symbols) 2 - Exponents - Multiplication / Division (left to right) 4 - Addition/Subtraction (left to right) Find each answer ( - 7) = =. 2 + ( - 25) = ( - 8) = 5. - ( - 5) = = 7. ( - 9)( - 8) = 8. (2)( - 2) = / - 7 = / + 8 =. ( - 2)( + 6)(- 5) = ( 2) = ( 8) 4 = 4. -( 8) + 2 = Answers:

2 Topic: Rationals page 4 Multiplying Fractions and Mixed Numbers ) Change any mixed numbers to improper fractions 2) Cross cancel any numerator with any denominator by dividing each by a common factor ) Multiply numerator by numerator and denominator by denominator 4) Simplify your answer (make it a mixed number if you can) Dividing Fractions and Mixed Numbers ) Change any mixed numbers to improper fractions 2) Remember Keep-Change-Flip: keep the first fraction, change the division sign to a multiplication sign, and flip the second fraction ) Multiply numerator by numerator and denominator by denominator 4) Simplify your answer (make it a mixed number if you can) Adding and Subtracting Fractions and Mixed Numbers ) Check to see if the denominators are the same; if not, find a common denominator 2) Now add or subtract the fractions remember, keep the denominator! ) Add or subtract the whole numbers 4) Simplify the fraction 5) Rewrite the sum or difference 5 2) 84 2 ) ) Answers:. 2. 4) ) 2 5 6) ) 7) )

3 Topic: Combining Like Terms and Applying the Distributive Property In algebraic expressions, like terms are terms that contain the same variables raised to the same power. Only the coefficients of like terms may be different. In order to combine like terms, we add or subtract the numerical coefficients of the like terms using the Distributive Property: ax + bx = (a + b)x. Examples:. 2x + 9x = (2 + 9) x = x 2. 2y - 7y = (2 7) y = 5y. 5x + 8-2x + 7 = x + 5 Here, the like terms are: 5x and -2x = x and: = 5 The Distributive Property of multiplication over addition/subtraction is frequently used in Algebra: Examples:. 7 (2x + 9) = 7 2x = 4x (6 5x) = 4 (6) - 4(5x) = 24-20x Simplify each expression by combining like terms.. 8y + 2y y + 4y + 9 =. x + 7 2x = 4. 8n 7y 2n + 5 y = Answers: Apply the distributive property and write your answer in simplest form. 5. 7(x 4) = 6. 5(4n ) = 7. -6(y + 5) = 8. -4(8 9x) = (n + 7) 0n = (5 + 7y) + 6(2y 9) = 0.

4 Topic: Algebra Solving equations by using the Addition, Subtraction or Multiplication Property of Equality. Check the solution. Ex : x Ex 2: 7x 6 - x = x 6 x x 4x 6 4 x Check: x 4x x = 2 ( ) = 9 9 = 9 Check: 7x x = - 4 Translate and evaluate the following equations. Ex : The product of 4 and a number is n 28 4n n 7 Addition: sum, more than Multiplication: product 7(2) - 6 -(2) = = = -4-4 = -4 Ex 4. The quotient of a number and is 5. n 5 n 45 Subtraction: difference, less than Division: quotient Solve the following equations. Show your work and check your solution.. 2x x. 5x + 8 = - 2 Check: Check: Check: 4

5 Apply the distributive property first. 4. 4x x (x 7) = 8 4 Check: Check: Check: Apply the distributive property first. 7. 8x 5 6x = 7 8. = 4x 0x x ( + 8x) = - Check: Check: Check: Translate each sentence to an algebraic equation. Then use mental math to find the solution. Equation Solution 0. One-half of a number is more than 7 times a number is less than three times a number is increased by twice a number is 24. 5

6 Topic: Angle Relationships and Algebra Notation: m means the" measure of angle" means congruent or equal in measure Vertical Angles Complementary Angles Supplementary Angles Angles that are opposite each other across two intersecting lines. m m and m 2 m 4 Two angles whose sum is 90 o. m m Two angles whose sum is 80 o. m m State how the angle labeled x is related to the angle with the given measurement. Find the value of x in each figure. ) 49º x º ) x = 2) 2) x º x = 62º ) 6 xº 2º ) x =

7 4) Find the missing angles. Note: the angles are not drawn to scale. Given: 4 = 50 Find each angle and write your reasoning. = = = 5) 5) (x - 4) 2 (x + 0) Relationship x = m = m 2 = m = 7

8 Topic: Geometry You should know the following formulas and be able to use them to find the area or perimeter of a geometric figure. Perimeter of a polygon = the sum of the sides Rectangle: P = 2l + 2w A = lw Square: P = 4s A = s 2 Parallelogram: P = s + s2 + s + s4 A = bh Triangle: P = s + s2 + s A = ½ bh Trapezoid: P = s + s2 + s + s4 A = ½ (b + b2)h Circle: Circumference = πd A = πr 2 Find the perimeter/circumference and area of each figure. Express #6 in terms of pi (π). Show your work. (Use and attach a separate work page if space is needed.) cm 4.5 cm 7 cm cm 8 cm 20 cm 5 in 4 in 9 in 7 ½ in 5. 5 cm 6. 5 ft 5 cm 4 cm 5 cm cm 8

9 9) Name each figure. Find the volume or surface area of each. (Use the reference sheet at the back!) a) Name: b) Name: Volume: Surface Area: (in terms of pi) 2 in. 4 in. 0 in. 0 in. 5 in. c) Name: d) Name: Volume: Volume: (in terms of pi) in. in. 9 in. 0) A storage tank shaped like a rectangular prism is being manufactured to hold 00,000 cubic feet of natural gas. It has a length of 0 feet and a width of 25 feet. Use algebra to find out what height the tank should be. 9

10 Topic: Ratio & Proprtion ) Buck drove 220 miles in 5 hours. What was his average rate of speed? 2) Horace read 60 pages in 4 hours. How many pages can he read in 6 hours? ) Pasha bought pounds of onions for $2.67. Which ratio is proportional to pounds at $2.67? A. B. C. D. 0

11 8 th Grade Summer Math Packet Unit: Knowledge of Algebra, Patterns, and Functions Objective: Write an algebraic expression to represent unknown quantities with one unknown and or 2 operations. Examples: The tables below show phrases written as mathematical expressions. Phrases Expression Phrases Expression 9 more than a number the sum of 9 and a number a number plus 9 X subtracted from a number a number minus 4 4 less than a number h - 4 a number increased by 9 a number decreased by 4 the total of x and 9 the difference of h and 4 Phrases Expression Phrases Expression 6 multiplied by g 6 times a number the product of g and 6 6g a number divided by 5 t the quotient of t and 5 divide a number by 5 5 Write each phrase as an algebraic expression..) 7 less than m 2.) The quotient of and y.) 7 years younger than Jessica 4.) times as many marbles as Bob has 5.) Let t = the number of tomatoes Tye planted last year. This year she planted times as many. Write an algebraic expression to show how many tomatoes Tye planted this year. 6.) Last week Jason sold x number of hot dogs at the football game. This week he sold twice as many as last week, and then he sold 0 more. Write an expression to show how many hot dogs Jason sold this week.

12 8 th Grade Summer Math Packet Unit: Knowledge of Algebra, Patterns, and Functions Objective: Evaluate an algebraic expression using one unknown and no more than 2 operations. Example : Evaluate 6x 7 if x = 8. Example 2: Evaluate 5m 5 if m = 6. 6x 7 = 6(8) 7 Replace x with 8. 5m 5 = 5(6) 5 Replace m with 6.. = 48 7 Use order of operations. = 0 5 Use order of operations. = 4 Subtract 7 from 48. = 5 Subtract 5 from 0. Example : Evaluate 7b ( 7 ) ( 6 ) = 42 = 7b if b = 6. Example 4: Evaluate x + 4 if x =. Replace b with 6. x + 4 = + 4 Replace x with. Multiply 6 by 7. = Use order of operations. = 4 Divide. = Add 27 and 4. Evaluate the following expressions using the given values for a, b, and c. Show each step!.) Evaluate 6 + b if b = 7 2.) Evaluate 6a 2 if a = 4.) Evaluate 5(6) c if c = 7 4.) Evaluate b 4 if b = ) Evaluate 7. 5 m 6.) Evaluate (n ) 2 if n = 9 if m =

13 8 th Grade Summer Math Packet Unit: Knowledge of Algebra, Patterns, and Functions Objective: Evaluate numeric expressions using order of operations with no more than 4 operations. Use the order of operations to evaluate numerical expressions.. Do all operations within grouping symbols first. 2. Evaluate all powers before other operations.. Multiply and divide in order from left to right. 4. Add and subtract in order from left to right. Example : Evaluate 4 + (7 2) 2! 5 Example 2: 8 + ( + 5) (7 2) 2! ( + 5) 2 4 = 4 + (5) 2! 5 Subtract first since 7 2 is in parentheses = 8 + (6) 2 4 Add first since + 5 is in parentheses = ! 5 Multiply left to right,! 5 = 5 = Find the value of 6 2 = Multiply left to right, 2! 5 = 0 = Divide 6 by 4 = 29 0 Add left to right, = 29 = 7 Add 8 and 9 = 9 Subtract 0 from 29 Evaluate each of the following. Show each step!.) (2 + 0) ) (6 + 5)! (8 6).) 72 5(2.8) )! 4(0 8) 60 5.) The perimeter of a hexagon is found by adding the lengths of all six sides of the hexagon. For the hexagon below write a numerical expression to find the perimeter. Then evaluate the expression. 6.) Without parentheses, the expression equals 27. Place parentheses in the expression so that it equals ; then 2. 5

14 8 th Grade Summer Math Packet Unit: Knowledge of Algebra, Patterns, and Functions Objective: Determine the unknown in a linear equation with or 2 operations Remember, equations must always remain balanced. If you add or subtract the same number from each side of an equation, the two sides remain equal. If you multiply or divide the same number from each side of an equation, the two sides remain equal. Example : Solve x + 5 = x + 5 = Write the equation x + 5 = Write the equation - 5 = - 5 Subtract 5 from both sides Check = Replace x with 6 x = 6 Simplify =! The sentence is true Example 2: Solve - 2 = - y - 2 = - y Write the equation - 2 = - y Write the equation - = - Divide each side by Check - 2 = - (7) Replace the y with 7 7 = y Simplify -2 = - 2? Multiply is the sentence true? Example : Solve x + 2 = 2 x + 2 = 2 Write the equation x + 2 = 2 Write the equation - 2 = - 2 Subtract 2 from each side (7) + 2 = 2? Replace x with 7 x = 2 Simplify Check = 2? Multiply Divide each side by 2 = 2? Add is the sentence true? x = 7 Simplify.) Solve x 9 = -2 2.) Solve 48 = - 6r.) Solve 2t + 7 = - 4.) Solve 4t +.5 = ) It costs $2 to attend a golf clinic with a local pro. Buckets of balls for practice during the clinic cost $ each. How many buckets can you buy at the clinic if you have $0 to spend? 6.) An online retailer charges $6.99 plus $0.55 per pound to ship electronics purchases. How many pounds is a DVD player for which the shipping charge is $.94? 8 4

15 Unit: Knowledge of Algebra, Patterns, and Functions Objective: Graph rational numbers on a number line. 8 th Grade Summer Math Packet Rational Numbers are numbers that can be written as fractions. Some examples of rational numbers are ½, 5 ¾, 0.8, and Example: Graph and label the following numbers on the number line: A: 2 B: 4 4 C: D: 2.5 C A D B " " " ".) Graph and label the following numbers on the number line. 2.) Graph and label the following numbers on the number line. A: 5 B: C: 2 D: 5 A: 0 B: 2 C: 5 2 D: 4.) Graph and label the following numbers on the number line. 4.) Graph and label the following numbers on the number line. A:.5 B: 0.5 C:.5 D:.5 A: 9 B: 2 C: 9 4 D: 2 5.) Jonah recorded the temperature for 5 days on a chart. Draw a number line and graph the temperatures. Where do the numbers on the line need to begin and end? Label the points to 5. Day Day 2 Day Day 4 Day ) Graphing numbers on a number line can help you put them in order from smallest to greatest. Draw a number line and graph the numbers in the chart below. Label the points. Which number is the smallest? V W X Y Z

16 Unit: Knowledge of Algebra, Patterns, and Functions Objective: Graph ordered pairs in a coordinate plane. 8 th Grade Summer Math Packet The coordinate plane is used to locate points. The horizontal number line is the x-axis. The vertical number line is the y-axis. Their intersection is the origin. Points are located using ordered pairs. The first number in an ordered pair is the x-coordinate; the second number is the y-coordinate. The coordinate plane is separated into four sections called quadrants. Example : Name the ordered pair for point P. Then identify the quadrant in which P lies. Start at the origin. Move 4 units left along the x-axis. Move units up on the y-axis. The ordered pair for point P is (- 4, ). P is in the upper left quadrant or quadrant II. Quadrant 2 Quadrant Example 2: Graph and label the point M (0, - 4). Start at the origin. Move 0 units along the x-axis. Move 4 units down on the y-axis. Draw a dot and label it M(0, - 4)..) Name the ordered pair for each point graphed at the right. Then identify the quadrant in which each point lies. Quadrant Quadrant 4 2.) Find each of the points below on the coordinate plane. Then identify the quadrant in which each point lies. Coordinates Quadrant Coordinates Quadrant P (, ) A (, ) Q (, ) J (, ) R (, ) B (, ) S (, ).) Graph and label each point on the coordinate plane. N (, -) P (-2, 4) Q (-, -4) R (0, 0) S (-5, 0) H (, ) 4.) Graph and label each point on the coordinate plane. D (0, 4) E (5, 5) G (-, 0) H (-6, -2) J (0, -2) 6

17 Unit: Knowledge of Geometry Objective: Determine the congruent parts of polygons. 8 th Grade Summer Math Packet Congruent Polygons Non Congruent Polygons Congruent Polygons Congruent Segments Congruent Angles Corresponding Sides of a Polygon Corresponding Angles of a Polygon ABC DEF A D cm cm B 7cm C E 7cm F Polygons that have exactly the same size and the same shape Segments that have the same length Angles that have the same measure Sides of a polygon that are matched up with sides of another congruent or similar polygon Angles of a polygon that match up with angles of another congruent or similar polygon Corresponding sides and angles of congruent polygons are congruent: AB DE BC EF AC DF A D B E C F.) G 5cm H 4cm F 45 8cm J K z L 8cm y M x N 2.) Use the figures in problem # to complete the following congruence statements. G K Polygon FGHJ polygon NMLK Complete the following congruence statements. H F GH KL IJ.) Look at the figures in problem #. Determine the measure of each segment or angle. 4.) Polygon HJKLMNPQ is congruent to polygon RSTUVXYZ. What is the length, in units, of RZ? (Note: Figures are not drawn to scale.) x = y = z = 9 7

18 8 th Grade Summer Math Packet Unit: Knowledge of Number Relationships & Computation Objective: Determine equivalent forms of rational numbers expressed as fractions, decimals, percents, and ratios. - A Examples: To write a decimal as a fraction, divide the numerator of the fraction by the denominator. Use a power of ten in the denominator to change a decimal to a fraction Write as a decimal = 0. 5 because 5 repeats forever Write 0.2 as a fraction in simplest form. 0.2 = = = ) Write using bar notation to represent the repeating decimal. 2.) Write 5 as a decimal..) Write 5 4 as a decimal. 8 4.) Write 0.94 as a fraction in simplest form. 5.) Write 0.48 as a fraction in simplest form. 6.) There were 6 girls and 8 boys in Mrs. Johnson s math class. Write a ratio of the # of girls to the # of boys in fraction form. Then write the fraction as a repeating decimal. 8 8

19 8 th Grade Summer Math Packet Unit: Knowledge of Number Relationships & Computation Objective: Determine equivalent forms of rational numbers expressed as fractions, decimals, percents, and ratios.- B Examples: A RATIO is a comparison of two numbers by division. When a ratio compares a number to 00, it can be written as a PERCENT. To write a ratio or fraction as a percent, find an equivalent fraction with a denominator of 00. You can also use the meaning of percent to change percents to fractions. 9 Write as a percent = = 95% Since = 5, multiply the numerator and denominator by 5. Write 92% as a fraction in simplest form = = Write 92% as a decimal. Move decimal two places to the left. Add zeros if needed. 92.0% = 0.92 Write 0.4 as a percent. Move decimal two places to the right. Add zeros if needed. 0.4 = 40% 7 2.) Write 9% as a decimal and fraction in simplest form..) Write as a percent and decimal ) Write 75% as a decimal and fraction in simplest form..) Write as a percent and decimal ) Ms. Crest surveyed her class and found that 5 out of 0 students brushed their teeth more than twice a day. Write this ratio as a fraction in simplest form, then write it as a % and a decimal. 6.) A local retail store was having a sale and offered all their merchandise as a 25% discount. Write this percent as a fraction in simplest form, then write it as a decimal. 9 9

20 8 th Grade Summer Math Packet Unit: Knowledge of Number Relationships & Computation Objective: Compare, order, and describe rational numbers. Examples: RATIONAL numbers include fractions, decimal, and percents. To COMPARE or ORDER rational numbers, they must be in the same form (all fraction or all decimals, or all %s) Example: Order 0.6, 48%, and 2 from least to greatest. Step Change all to decimals % = 0.48 Step 2 Compare decimals & Order. 0.48, 0.5, 0.6 = Step Write using original form. 48%,, ) Order from least to greatest. 22%, 0., 5 2.) Order from least to greatest. 0.74, 4, 70%.) Replace with <, >, or = % 4.) Which is the largest? ) According to the Pet Food Manufacturer s Association, out of 25 people own large dogs and out of 50 medium dogs. Do more people own large or medium dogs? 6.) Your PE teacher asked you to run for specific time period. You ran 0.6 of the time. Two of your friends ran 7 and 72% of the time. Order the amount of time you 0 and your friends ran from least to greatest

21 8 th Grade Summer Math Packet Unit: Knowledge of Number Relationships & Computation Objective: Add, subtract, multiply and divide integers. - A Examples: ADDITION INTEGER RULES: For integers with the same sign: The sum of two positive integers is POSITIVE. The sum of two negative integers is NEGATIVE. For integers with different signs, subtract their absolute value. The sum is: Positive IF the positive integer has the greater absolute value. Negative IF the negative integers has the greater absolute value. Examples: (- ) = add keep the sign = (- 2) = add keep the sign = (- 7) = subtract keep the sign of the higher = = subtract keep the sign of the higher = - SUBTRACTION INTEGER RULES: Keep the first number the same Switch the subtraction sign to ADDITION Change the second number to it s opposite. Opposite: - 6 to 6 Follow Addition rules above. Examples: 6 9 = 6 + (- 9) = (- 2) = = 2-7 = - + (- 7) = - 0 ( - 2) = + 2 =.) Add: 2 + (- 7) 2.) Subtract: ) Evaluate a b if a = - 2 and b = ) Evaluate x + y + z if x =, y = - 5, and z = ) In Mongolia the temperature can dip down to 45 o C in January. The temperature in July may reach 40 o C. What is the temperature range in Mongolia? 6.) Write an addition expression to describe skateboarding situation. Then determine the sum. Hank starts at the bottom of a half pipe 6 feet below street level. He rises 4 feet at the top of his kickturn. 4 2

22 8 th Grade Summer Math Packet Unit: Knowledge of Number Relationships & Computation Objective: Add, subtract, multiply and divide integers. - B Examples: MULTIPLYING & DIVIDING INTEGER RULES: Two integers with DIFFERENT signs the answer is NEGATIVE. Two integers with SAME signs the answer is POSITIVE. Examples: 5 (- 2) = 5 times 2, the signs are different so the answer will be negative = - 0 (- 6) (- 9) = the signs are the same so the answer will be positive = 54 0 (- 5) = the signs are different so the answer will be negative = (- 5) = the signs are the same so the answer will be positive = 20.) Mulitply: - 4 (- 7) 2.) Divide: 50 (- 25).) Evaluate if a = - and c = 5 4.) Evaluate if d = - 24, e = - 4, and f = 8 - ac de f 5.) A computer stock decreased 2 points each hour for 6 hours. Determine the total change in the stock value over the 6 hours. 6.) A submarine descends at a rate of 60 feet each minute. How long will it take it to descend to a depth of 660 feet below the surface? 42 22

23 8 th Grade Summer Math Packet Unit: Knowledge of Number Relationships & Computation Objective: Add, subtract, and multiply positive fractions and mixed numbers. - A Examples: To add unlike fractions (fractions with different denominators), rename the fractions so there is a common denominator. 2 Add: + = x5 = = 6 x x6 = = 5 x = Add: = 2 2 x = x = x2 = 8 x2 = = is improper so we must change it to proper. 7 divided by 6 = = ) Add: ) Add: ) Add: ) Add: ) A quiche recipe calls for A recipe for quesadillas requires 2 cups of grated cheese. 4 cups of grated cheese. What is the total amount of grated cheese needed for both recipes? 6.) You want to make a scarf and matching hat. The 7 pattern calls for yards of fabric for the scarf and 8 2 yards of fabric for the hat. How much fabric do you 2 need in all? 4 2

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Topic: Integers. Addition Subtraction Multiplication Division Same signs: Add & keep sign = = - 10.

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