SUMMER REVIEW PACKET 2 FOR STUDENTS ENTERING ALGEBRA 1

Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN

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SUMMER REVIEW PACKET 2

FOR STUDENTS ENTERING ALGEBRA 1

Dear Students,

Welcome to Ma’ayanot. We are very happy that you will be with us in the Fall.

The Math department is looking forward to working with you and watching you learn. Your teachers are very supportive and want you to understand and enjoy learning math.

This packet is designed to help you to review the skills that are essential as a foundation for your algebra class. There are model problems which will help to guide you.

INSTRUCTIONS:

1) PRINT OUT PACKET

2) SHOW YOUR WORK IN THE SPACE PROVIDED IN THE PACKET.

3) NO CALCULATOR

4) FOR THE FRACTION PROBLEMS STARTING (p.11-16) do the ODD problems only.

5) HAND YOUR PACKET TO YOUR TEACHER ON THE FIRST DAY OF SCHOOL!

GOOD LUCK AND HAVE A GREAT SUMMER!

THE MATH DEPARTMENT


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Objective: Write an algebraic expression to represent unknown quantities. • A variable is a symbol, usually a letter, used to represent a number. • Algebraic expressions are combinations of variables, numbers, and at least one operation. Sum: addition Ex: x+ 5 is “the sum of a number and 5 “ or “5 added to a number” Difference : subtraction Ex: x-12 is “ the difference between a number and 12” or “12 less than a number” but 12 –x is “ the difference between 12 and a number” or “12 decreased by a number” Product: multiplication Ex: 20x is “the product of 20 and a number”

Quotient: Division Ex: 20

x is “the quotient of 20 and a number” or “ 20 divided by a number”

But 20

x is “ quotient of a number and 20” or “ a number divided by 20”

In each case, let x= the number. Be careful with difference and quotient!

1.) a number plus 2

1

2.) the sum of a number and 73

3.) the difference of 21 and a number

4.) the difference of a number and 21

5) a) Five less than a number

6.) Five decreased by a number

7) The quotient of 20 and a number

8) Twenty divided by a number

9) The quotient of a number and 20

10) A number divided by 20

10. John is 5 years younger than Bill; If john’s age is x, what is Bill’s age in terms of x

10. Bill is 5 years younger than John. If John’s age is x, what is bill’s age in terms of x?


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Objective: Evaluate an algebraic expression. • A variable is a symbol, usually a letter, used to represent a number. • Algebraic expressions are combinations of variables, numbers, and at least one operation. • Multiplication in algebra can be shown as 4n or 4 x n • The variables in an algebraic expression can be replaced with any number. • Once the variables have been replaced, you can evaluate, or find the value of, the algebraic expression. Examples: Evaluate 44 + n if n= 9 44 + n original expression 44 + 9 replace the variable with it’s value 53 solution

1.) Evaluate 150 + n if n = 15 2.) Evaluate 12n if n = 9

3.) Evaluate 15n + 19 if n = 3

1

4.) Evaluate 30n if n = 2.5

5.)Evaluate 24n k if n = 6 and k = 8

6.)Evaluate nk – 2b + 8 if b = 1.5, k = 8, and n = 7


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Unit: KNOWLEDGE of ALGEBRA, PATTERNS, and FUNCTIONS Objective: Evaluate numeric expressions using order of operations. • A numerical expression is a combination of numbers and operations. • The Order of Operations tells you which operation to perform first so that everyone gets the same final answer. • The Order of Operations is: Parentheses, Exponents, Multiplication or Division (left to right), and Addition or

Subtraction (left to right.) Examples:

48 (3 + 3) – 22 original expression

48 6 - 22 P simplify the expression inside the parentheses

48 6 – 4 E exponents 8 – 4 M,D multiplication and division from left to right 4 A, S addition and subtraction from left to right Show your steps. No calculator!

1.) (8 + 1) x 12 – 13

2.) 13 x 4 – 72 8

3.) 88 – 16 x 5 + 2 – 3

4.) 100 52 x 43

5.) 45 9 – 3 + 2 x 3

6.) (52 + 33) x (81 + 9) 10


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Unit: solve 1 step equation with addition and subtraction Objective: Determine the unknown in a linear equation (addition & subtraction). • Addition equations: Subtract the same number from each side of the equation so that the two sides remain equal. • Subtraction equations: Add the same number to each side of the equation so that the two sides remain equal. Examples: b + 3 = 6 original equation b – 8 = 4 original equation - 3 - 3 subtract 3 from each side +8 +8 add 4 to each side b + 0 = 3 solution b + 0 = 12 solution b = 3 simplify b = 12 simplify

1.)

g + 5 = 12

2.)

s – 12 = 29

3.)

m + 3.5 = 10.5

4.)

k – 5.5 = 8.5

5.)

w + 6.25 = 22

6.)

g – 3.75 = 49.75


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Unit: Solve 1 step equation with multiplication and division Objective: Determine the unknown in a linear equation (multiplication & division). • In a multiplication equation, the number by which a variable is multiplied is called the coefficient. In the multiplication

equation 2x = 8, the coefficient is 2. • Multiplication equations: Divide both sides by the coefficient so that the two sides remain equal.

• In a division equation, the number by which the variable is divided is called the divisor. In the division equation 4

x,

4 is the divisor. • Division equations: Multiply both sides of the equation by the divisor so that the two sides remain equal. Examples:

4b = 16 original equation 6

m = 11 original equation

4 4 divide both sides by 4 6 x 6

m = 11 x 6 multiply each side by 6

1b = 4 solution 1m = 66 solution b = 4 simplify m = 66 simplify

1.) 7x = 63

2.)

9

k = 8

3.)

5b = 3.55

4.)

7

n = 5.55

5.)

12m = 84.72

6.)

13

p = 2.67


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Unit: NUMBER RELATIONSHIPS and COMPUTATION Objective: Read, write, and represent integers. Examples: Integer: Any number from the set {… -3,-2,-1,0,1,2,3…} Write an integer to describe each situation EX: a height increase of 3 inches The word increase represents positive. Answer: 3 or +3. EX: 50 feet below sea level The word below represents negative. Answer: - 50

1.) Write an integer to describe: The stock market increased 75 points

2.) Write an integer to describe: A loss of 15 yards

3.) Write an integer to describe the situation: Nancy owes her friend $10

4.) Write an integer to describe: Frederick is located 290 feet above sea level.

5.) Write an integer to describe: The temperature was 3° below zero

6.) Write an integer to describe: The 6th grade has 12 fewer students than last year

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Integers less than zero

are negative integers Integers greater than zero

are positive integers

Negative integers are written with a - sign

Positive integers can be written with or without a + sign Zero is neither nor positive


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FRACTIONS- Methods

A. Changing a Mixed Number to an Improper Fraction

Mixed number – 4 3

2 (contains a whole number and a fraction)

Improper fraction - 3

14 (numerator is larger than denominator)

Step 1 – Multiply the denominator and the whole number

Step 2 – Add this answer to the numerator; this becomes the new numerator

Step 3 – Carry the original denominator over

Example #1: 3 8

1 = 3 × 8 + 1 = 25

8

25

Example #2: 4 9

4 = 4 × 9 + 4 = 40

9

40

B. Changing an Improper Fraction to a Mixed Number

Step 1 – Divide the numerator by the denominator

Step 2 – The answer from step 1 becomes the whole number

Step 3 – The remainder becomes the new numerator

Step 4 – The original denominator carries over

Example 5

47 = 47 ÷ 5 or 5 47 = 9, remainder 2

Ans: 2

95


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C. Reducing Fractions

Step 1 – Find a number that will divide into both the numerator and the

denominator

Step 2 – Divide numerator and denominator by this number

Example #1: 15

10 =

3

2 (because both 10 and 15 are divisible by 5)

Example #2: 8

4 =

2

1 (because both 4 and 8 are divisible by 4)

D. Raising Fractions to Higher Terms When a New Denominator is Known

Step 1 – Divide the new denominator by the old denominator

Step 2 – Multiply the numerator by the answer from step 1 to find the new

numerator

*Note: If the original number is a mixed number, convert it to an improper

fraction before raising to higher terms (see Example #2)

Example #1: 3

2 =

12 becomes

3

2 =

12

8 because 12 ÷ 3 = 4

and 2 × 4 = 8

Example #2: 2 5

1 =

20 becomes

5

11 =

20 becomes

5

11 =

20

44

because 20 ÷ 5 = 4 and 11 × 4 = 44

E. (1) Multiplying Simple Fractions

Step 1 – Multiply the numerators

Step 2 – Multiply the denominators

Step 3 – Reduce the answer to lowest terms

Example: 7

1 ×

6

4 =

42

4 which reduces to

21

2


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E. (2) Multiplying Mixed Numbers

Step 1 – Convert the mixed numbers to improper fractions first

Step 2 – Multiply the numerators


Step 4 – Reduce the answer to lowest terms

Example: 23

1 × 1

2

1 =

3

7 ×

2

3 =

6

21 which then reduces to 3

2

1

F (1) Dividing Simple Fractions

Step 1 – Change division sign to multiplication

Step 2 – Change the fraction following the multiplication sign to its

reciprocal (flip the fraction around so the old denominator is the

new numerator and the old numerator is the new denominator)

Step 3 - Multiply the numerators


Step 5 – Change the answer to lowest terms

Example: 8

1 ÷

3

2 = becomes

8

1 ×

2

3 which when solved is

16

3

F(2). Dividing Mixed Numbers

Step 1 – Convert the mixed number or numbers to improper fraction

Step 2 – Change the division sign to multiplication

Step 3 – Change the fraction following the multiplication sign to its

reciprocal (flip the fraction around so the old denominator is the

new numerator and the old numerator is the new denominator)

Step 4 - Multiply the numerators



Example: 34

3 ÷ 2

6

5 = becomes

4

15 ÷

6

17 becomes

4

15 ×

17

6 =

which when solved is 24

15 ×

17

63 =

34

45 which simplifies to 1

34

11


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G. Adding and Subtracting Fractions

Step 1 – Find a common denominator (a number that both denominators

will go into)

Step 2 – Raise each fraction to higher terms as needed

Step 3 – Add or subtract the numerators only as shown

Step 4 – Carry denominator over


Example #1: 2

1 +

8

7 = Common denominator is 8 because both 2 and

8 will go into 8

2

1 =

8

4

+ 8

7 =

8

7

8

11 which simplifies to 1

8

3

Example #2: 45

3 –

4

1 = Common denominator is 20 because both 4

and 5 will go into 20

45

3 = 4

20

12

– 4

1 =

20

5

420

7

Example #3: 2 8

1 = 2

8

1 = 12

8

1+

8

8 = 1

8

9

– 14

1 = 1

8

2 = 1

8

2 = 1

8

2

8

7 **


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INSTRUCTIONS: YOU SHOULD DO THE ODD PROBLEMS ONLY IN THIS SECTION ( See previous pages for instructions on how to do each type of problem)

A. Write as an improper fraction.

1. 18

1 2. 4

5

1 3. 1

3

2 4. 2

16

3

5. 27

5 6. 2

16

1 7. 1

8

5 8. 3

5

4

9. 74

1 10. 5

3

2 11. 3

6

5 12. 6

2

1

B. Write as a mixed number.

1. 4

10 2.

2

19 3.

3

25 4.

8

9

5. 16

25 6.

4

35 7.

3

7 8.

8

21


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C. Write in lowest terms.

1. 32

6 2.

35

21 3.

24

18 4.

15

12

5. 30

5 6.

27

9 7.

49

14 8.

32

8

9.. 121

12 10. 2

20

16 11. 5

14

8 12. 3

25

10


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D. Find the missing numerator by raising the fraction to higher terms.

1. 4

3 =

12

? 2.

16

7 =

64

? 3.

8

5 =

48

? 4.

9

5 =

72

?

5. 53

2 =

12

? 6. 1

5

4 =

10

? 7. 1

4

1 =

12

? 8. 2

5

3 =

10

?


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E. Multiply.

1. 9

1 ×

2

1 = 2.

10

7 ×

5

2 = 3.

8

3 ×

7

2 = 4.

2

1 ×

16

3 =

5. 4

3 ×

3

2 = 6.

16

7 ×

3

4 = 7.

64

15 ×

12

1 = 8.

9

2 ×

9

5 =

9. 4

3 × 10 = 10. 1

2

1 ×

6

5 = 11.

16

3 ×

12

5 = 12. 14 ×

8

3 =

13. 2

1 × 1

3

1 = 14. 3

16

1 ×

5

1 = 15. 18 × 1

2

1 = 16. 16 × 2

8

1 =


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17. 68

3 × 1

5

3 = 18. 2

3

2 × 4

8

3 = 19. 4

9

4 × 4

4

2 = 20. 3

8

1 × 2

5

2=

F. Divide as shown.

1. 2

1 ÷

4

1 = 2.

5

2 ÷

2

1 = 3.

3

8 ÷

3

2 = 4.

9

2 ÷

3

1 =

5. 4 ÷ 8

1 = 6. 8 ÷

5

4 = 7. 9 ÷

4

3 = 8.

5

6 ÷

5

4 =

9. 11

4 ÷

11

1 = 10.

7

2 ÷

9

5 = 11.

3

2 ÷ 4 = 12. 14 ÷

8

7 =


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13. 15 ÷ 6

5 = 14. 8 ÷

4

3 = 15. 1

4

1 ÷ 1

2

1 = 16. 3

2

1 ÷ 5 =

17. 64

1 ÷ 2

2

1 18. 5

3

1 ÷ 2

3

2 = 19. 2

4

3 ÷ 1

8

1 = 20. 3

5

1 ÷ 1

7

5=

G. Add or subtract as shown.

1. 8

3 +

8

7 = 2.

3

2 +

4

3 = 3.

32

3 +

8

1 = 4.

5

3 +

6

5 =

5. 8

5 +

10

1 = 6.

8

3 + 1

4

1 = 7.

4

1 +

5

1 = 8. 2

8

1 + 1

4

1=


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9. 18

5 +

16

13 = 10. 2

3

2 +

9

4 = 11.

10

9 –

16

3 = 12.

8

7 –

2

1 =

13. 16

11 –

4

1 = 14.

6

5 –

5

1 = 15.

8

7 –

10

3 = 16. 1

2

1 –

32

3 =

17. 56

5 – 2

9

3 = 18. 3

3

2 – 1

8

7 = 19. 2

4

1 –

6

5 = 20. 4

6

5 – 1

2

1=


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Converting between decimals and fractions and percents: please read this carefully

There are many different methods to convert between decimals, fractions and percents, so we must understand the meaning

1. Percent (%) means per hundred

so 52% = 52/100 which means as a fraction it is 52

100 which reduces to

13

25

and as a decimal it is 52 100 .52

2. A) To multiply by 100, move decimal point 2 places to the right

To divide by 100, we can move the decimal point over 2 places to the left

So .52 x 100 = 52 .735 x 100 = 73.5 12.3 x 100 = 123

52/ 100 = .52 73.5/ 100 = .735 123/100 = 12.3

B) To multiply by, decimal point right 1 digit

To divide by 10, move left by 1 digit

.3 x 10 = 3 .35 x 10 = 3.5 27.2 x 10 = 272

3/10 = .3 3.5/10= .35 272/10= 2.72

See next page!


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3. Conversion between fractions and decimals

A. From fraction to decimal : divide numerator by denominator

Ex: 3

3 5 .65

B. From decimal to fraction: write as a fraction using the place value of the last digit, then simplify if possible

Ex: 41

.41100

4 2

.410 5

279

.2791000

3

.03100

4. Conversion between decimals and percents

A. From percent to decimal

Since percent means per hundred, it means divided by 100

So 23

23% .23100

(remember there’s an easy way to divide by 100)

5.2

5.2% .052100

B. From decimal to percent:

You can think of this as the reverse of the above, multiply by 100

So . .23 23% .512 51.2%

There's more on next page- read it all carefully!


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5) Conversion between fraction and percents

A. From percent to fraction, since percent means per hundred , write the number over 100 and simplify the fraction:

2323%

100

55 1155%

100 20

14.214.2%

100 but we don’t want a decimal in the fraction so change to

14.2 10 14214.2%

100 10 1000 which can be simplified to

71

50

B) From fraction to percent, we can either set up a proportion

2

5 100

x and cross multiply 2 100 5x , and x=40, so it’s 40%

Or we can convert to a decimal first and change to a fraction

22 5 .4

5 and multiply by 100 to get 40%


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Do the following and show work: Please do all problems on this page

1 Write these decimals as fractions:

0.3 = …………. 0.5 = …………. 0.6 = …………. 0.02 = ………….

0.05 = …………. 0.25 = …………. 0.36 = …………. 0.125 = ………….

2 Write these fractions as decimals:

107 = ………………. 5

1 = ………………. 52 = ………………. 4

3 = …………………

87 = ………………. 3

2 = ………………. 209 =………………. 25

7 = …………………

3 Write these percentages as decimals:

3% = …………. 30% = …………. 25% = …………. 80% = ………….

8% = …………. 12% = …………. 67% = …………. 17.5% = ………….


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4 Write these percentages as fractions:

20% = …………. 75% = …………. 5% = …………. 30% = ………….

40% = …………. 15% = …………. 24% = …………. 35% = ………….

5 Write these decimals as percentages:

0.25 = …………. 0.5 = …………. 0.7 = …………. 0.07 = ………….

0.45 = …………. 0.09 = …………. 0.4 = …………. 0.375 = ………….

6 Write these fractions as percentages:

101 = …………. 5

1 = …………. 109 = …………. 4

3 = ………….

54 = …………. 20

17 = …………. 31 = …………. 3

2 = ………….


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Complete this table.

Fraction Decimal Percentage

101

51

103

52

21

53

107

54

109

41

43


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Fill the gaps in the table.

YOU ARE NOT REQUIRED TO COMPLETE THIS PAGE, BUT PLEASE DO IF YOU HAVE TIME.

Percentage

Fraction Decimal

10%

0.2

103

40%

0.5

53

70%

0.8

109

25%


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Unit: NUMBER RELATIONSHIPS and COMPUTATION Objective: Identify and determine equivalent forms of proper fractions as decimals, percents, and ratios. Key Concept: Ratio: a comparison of two numbers A ratio can be written in 3 ways: a:b a to b or a b

EX: Write the ratio as a fraction simplest form: 4 wins to 6 losses

Since the ratio can be written as: 6

4 we can the simplify to

3

2 or 2:3 or 2 to 3

1.) Write the ratio as a fraction simplest form: 12 boys to 15 girls

2.) Write the ratio as a fraction simplest form: 20 books to 24 magazines

3.) Write the ratio as a fraction simplest form: 10 circles to 15 triangles

4.) Write the ratio as a fraction simplest form: 8 cups to 2 servings

5.) Write the ratio as a fraction simplest form: 50 cars to 100 trucks

6.) Write the ratio as a fraction simplest form: 9 pencils to 11 pens


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Objective: Compare and order fractions and decimals

A. Ordering fractions only: 1) rewrite each fraction as an equivalent fraction

using the LCD 2) Compare the numerators

EX: order the fractions 12

7;

8

3;

2

1 from least to greatest

1) LCD of 2, 8, and 12 is 24

2) 24

12

2

1

24

14

12

7

24

9

8

3

3) Comparing the numerators:

12

7

2

1

8

3

SHOW WORK

B. Ordering fractions and decimals: 1) Change the fractions to decimals 2) Compare the decimals

EX: order the numbers 0.3; ;8

3 and 0.38

from least to greatest

1) 375.08

3

2) Compare the decimals:

0.3 < 0.375 < 0.38

Therefore: 38.08

33.0

1.)

Order the fractions 4

3;

6

5;

3


2.)

Order the numbers 0.78; ;4

3 and 0. 8 from least to greatest

3.)

Order the fractions 6

5;

10

7;

5


4.)

Order the numbers ;10

3 ;5

1 and 0.25 from least to greatest

5.) Order the fractions 6

5;

9

5;

2


6.)

Which number has the greatest value? 0.94; ;20

19 or 25

24

40-

40

56

60

24

375.0

000.38


Page 28 of 29

Unit: NUMBER RELATIONSHIPS and COMPUTATION Objective: Determine 10, 20, 25, or 50 percent of a whole number. Example: Determine 25% of 40

Method 1: Write a proportion and solve

25

100 40

1

4 40

1 10

4 10 40

10

40 40

10

x

x

x

x

so x

Therefore 25% of 40 is 10.

Method 2: Change the percent to a decimal and multiply

25%= 0.25

0.25 X 40 = 10.00

Therefore 25% of 40 is 10.

1.) Determine 20% of 65.

Show using both methods Method 1: Method 2:

2.) Determine 50% of 120. Show using both methods Method 1: Method 2:



40 X 0.25 200 +800 10.00


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5.) 20% of the 250 students ate pizza for lunch. How many students ate pizza?

6.) Nadia saved 10% on her CD purchase. If the CD originally cost $24.90, how much did she save?

7) Juan answered 25

24questions correctly on his quiz.

What percent of the questions did he get correct?

8) 78% of the class completed their homework last night. What fraction of the class completed their homework?

YOU'RE DONE!

HAND IT IN TO YOUR TEACHER ON THE FIRST DAY OF CLASS

Documents

SUMMER REVIEW PACKET 2 FOR STUDENTS ENTERING ALGEBRA 1