Table of Contents -...

Daily Skill Builders: Pre-Algebra

ii© Mark Twain Media, Inc., Publishers

Table of Contents

Table of Contents

Introduction to the Teacher ........................................ivNCTM Standards Matrix for Grades 5–8 ................... 1Activity # SkillFractions and Mixed Numbers

1. Factors ............................................................. 32. Determining Factors......................................... 33. Divisibility ......................................................... 44. Prime and Composite Numbers ....................... 45. Prime Factorization .......................................... 56. Using Factoring to Find the Greatest Common

Factor ............................................................... 57. Using Prime Factorization to Find the Greatest

Common Factor ............................................... 68. Least Common Multiple ................................... 69. Equivalent Fractions......................................... 7

10. Simplifying Improper Fractions ........................ 711. Writing Whole Numbers as Fractions ............... 812. Writing Mixed Numbers as Fractions ............... 813. Writing Fractions in Lowest Terms ................... 914. Adding Fractions With Like Denominators ....... 915. Adding Fractions With Unlike Denominators .. 1016. Subtracting Fractions With Like

Denominators ................................................ 1017. Subtracting Fractions With Unlike

Denominators ................................................ 1118. Adding and Subtracting Fractions in Word

Problems ........................................................ 1119. Multiplying Fractions ...................................... 1220. Dividing Fractions .......................................... 1221. Finding a Fraction of a Whole Number .......... 1322. Canceling in Fractions ................................... 1323. Review Adding and Subtracting Fractions/

Test Taking ..................................................... 1424. Review Multiplying and Dividing Fractions/

Test Taking ..................................................... 14Decimals

25. Rounding Decimals ........................................ 1526. Changing Fractions to Decimals .................... 1527. Multiplying and Dividing by 10; 100;

1,000; etc., ..................................................... 1628. Scientific Notation .......................................... 1629. Adding and Subtracting Decimals .................. 1730. Multiplying Decimals ...................................... 1731. Dividing Decimals by Whole Numbers ........... 1832. Dividing Decimals by Decimals ...................... 1833. Review Decimals/Test Taking ......................... 1934. Review Decimals/Test Taking ......................... 19

Integers and Variable Expressions35. Integers and Absolute Value .......................... 2036. Comparing Numbers ...................................... 20

37. Ordering Numbers ......................................... 21 38. Adding Integers .............................................. 21 39. Subtracting Integers ....................................... 22 40. Multiplying Integers ........................................ 22 41. Dividing Integers ............................................ 23 42. Order of Operations ....................................... 23 43. Variable Expressions ..................................... 24 44. Evaluating Variable Expressions .................... 24 45. Exponents ...................................................... 25 46. Multiplying and Dividing Exponents ............... 25 47. Negative Exponents ....................................... 26 48. Order of Operations ....................................... 26 49. Writing Variable Expressions for Real World Problems ........................................................ 27 50. Properties of Numbers ................................... 27 51. Commutative Property ................................... 28 52. Associative Property ...................................... 28 53. Distributive Property ...................................... 29 54. Properties of Zero and One ........................... 29Equations and Inequalities 55. Combining Like Terms ................................... 30 56. Simplifying Variable Expressions ................... 30 57. Solving Equations by Subtracting .................. 31 58. Solving Equations by Adding ......................... 31 59. Solving Equations by Adding or Subtracting .. 32 60. Problem Solving With Equations .................... 32 61. Solving Equations by Dividing........................ 33 62. Solving Equations by Multiplying ................... 33 63. Solving Equations by Dividing or Multiplying . 34 64. Problem Solving Using Equations.................. 34 65. Solving Two-Step Equations .......................... 35 66. Solving Two-Step Equations .......................... 35 67. Problem Solving by Writing and Solving Equations ....................................................... 36 68. Problem Solving Using Equations/ Test Practice .................................................. 36 69. Simplifying and Solving Equations ................. 37 70. Simplifying and Solving Equations ................. 37 71. Formulas ........................................................ 38 72. Formulas ........................................................ 38 73. Inequalities ..................................................... 39 74. Graphing Inequalities ..................................... 39 75. Graphing Inequalities ..................................... 40 76. Solving Inequalities by Adding and Subtracting ..................................................... 40 77. Solving Inequalities by Dividing or Multiplying ...................................................... 41 78. Review Solving Inequalities/Test Taking ........ 41Graphing in the Coordinate Plane 79. Graphing in the Coordinate Plane ................. 42


iii© Mark Twain Media, Inc., Publishers

Table of Contents (cont.)

Table of Contents

80. Graphing in the Coordinate Plane ................. 4281. Equations With Two Variables ........................ 4382. Graphing Linear Equations ............................ 4383. Graphing Linear Equations ............................ 4484. Slope .............................................................. 4485. Finding Slope From a Graph .......................... 4586. Negative Slope and Zero Slope ..................... 4587. Finding Slope Using Two Points ..................... 4688. Graphing Linear Equations Using Slope ........ 4689. Finding x- and y-Intercepts ............................ 4790. Equations in Slope-Intercept Form ................ 4791. Writing an Equation for a Line ....................... 4892. Changing Equations to Slope-Intercept

Form .............................................................. 4893. Review Graphing Linear Equations/

Test Taking ..................................................... 4994. Review Graphing Linear Equations/

Test Taking ..................................................... 4995. Graphing Inequalities ..................................... 5096. Systems of Linear Equations ......................... 5097. Graphing Systems of Linear Equations ......... 5198. Systems of Linear Equations With No

Solution .......................................................... 51Ratios, Proportions, and Percents99. Ratios ............................................................. 52

100. Rates ............................................................. 52101. Equivalent Ratios ........................................... 53102. Proportions .................................................... 53103. Using Proportions .......................................... 54104. Changing Percents to Fractions ..................... 54105. Changing Percents to Decimals ..................... 55106. Changing Decimals to Percents ..................... 55107. Changing Fractions to Percents ..................... 56108. Calculating Percents From a Graph ............... 56109. Percent Problems .......................................... 57110. Percent Problems .......................................... 57111. Percent Problems .......................................... 58112. Word Problems With Percents ....................... 58113. Percent of Change ......................................... 59114. Review Percents/Test Taking ......................... 59Rational Numbers and Irrational Numbers115. Comparing Rational Numbers ....................... 60116. Adding Rational Numbers .............................. 60117. Subtracting Rational Numbers ....................... 61118. Multiplying Rational Numbers ........................ 61119. Dividing Rational Numbers ............................ 62120. Solving Equations With Rational Numbers .... 62121. Solving Equations With Rational Numbers .... 63122. Solving Equations With Rational Numbers .... 63123. Square Roots ................................................. 64124. Square Roots ................................................. 64

125. Adding and Subtracting Square Roots .......... 65126. Multiplying and Dividing Square Roots .......... 65127. Approximate Square Roots ............................ 66128. Square Roots and Rational Numbers ............ 66129. Review Square Roots/Test Taking ................. 67130. Review Square Roots/Test Taking ................. 67131. Square Roots and Equations ......................... 68132. The Pythagorean Theorem ............................ 68133. Using the Pythagorean Theorem ................... 69134. Square Roots and Formulas .......................... 69Polynomials135. Polynomials .................................................... 70136. Evaluating Polynomials .................................. 70137. Combining Like Terms in Polynomials ........... 71138. Adding Polynomials ....................................... 71139. Problem Solving With Addition of Polynomials .................................................... 72140. Subtracting Polynomials ................................ 72141. Multiplying Polynomials .................................. 73142. Powers of Monomials ..................................... 73143. Multiplying Polynomials by Monomials........... 74144. Evaluating Products of Polynomials and Monomials ..................................................... 74145. Dividing Polynomials by Monomials ............... 75146. Multiplying Binomials ..................................... 75147. Problem Solving Using Polynomials .............. 76148. Problem Solving Using Polynomials .............. 76149. Problem Solving With Polynomials ................ 77150. Problem Solving With Polynomials ................ 77151. Reviewing Adding and Subtracting Polynomials/ Test Taking ..................................................... 78152. Reviewing Multiplying and Dividing Polynomials/ Test Taking ..................................................... 78Probability and Odds153. Patterns and Sequences ............................... 79154. Patterns and Sequences ............................... 79155. The Counting Principle .................................. 80156. Factorials ....................................................... 80157. Probability ...................................................... 81158. Independent and Dependent Events ............. 81159. Computing Probabilities ................................. 82160. Odds .............................................................. 82161. Using a Sample to Make Predictions ............. 83162. Using a Sample to Make Decisions ............... 83

Answer Key .............................................................. 84


iv© Mark Twain Media, Inc., Publishers

Introduction to the Teacher

Introduction to the Teacher

Both the No Child Left Behind Act and standardized testing require students to meet certain proficiency standards. These Daily Skill Builders are designed to provide students with the opportunity to review or gain extra practice with the skills they are learning in their regularcurriculum. They were written with the NCTM National Standards in mind. A matrix correlating the activities with the standards they address is included on pages 1–2.

Suggestions for Use: Each activity page is divided into two reproducible sections that can be cut apart and used separately. Activities could be used in class as a warm-up, a review of atopic covered earlier in the year, as extra practice on a topic currently being studied, in a learn-ing center for review or extra practice, or as a homework assignment.

Organization: Activities are arranged by topic and skill level and are progressively more dif-ficult within each topic area. Activities are designed to cover most areas that are addressed inan average pre-algebra curriculum. The Table of Contents indentifies the skills that each activ-ity covers. Since standardized testing is an important component of education, review activities provide practice in standardized test-taking formats. This helps students become familiar and comfortable with the format and provides test-taking practice.

Topics Covered: Topics covered in the Daily Skill Builders: Pre-Algebra book include:

• Fractions and Mixed Numbers

• Decimals

• Integers and Variable Expressions

• Equations and Inequalities

• Graphing in the Coordinate Plane

• Ratios, Proportions, and Percents

• Rational Numbers and Irrational Numbers

• Polynomials

• Probability and Odds

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3© Mark Twain Media, Inc., Publishers

Fractions & Mixed Numbers


ACTIVITY 2 Determining Factors

For each number, find all of the factors.

Example: 20 1, 2, 4, 5, 10, 20

1. 24 2. 30

3. 18 4. 25

5. 36 6. 45

7. 40 8. 29

9. 21 10. 12

ACTIVITY 1 Factors

If a number divides evenly into another number,then it is a factor of that number.

Example: The numbers 1, 2, 3, 4, 6, and 12 will all go into 12 evenly, so they are factors of 12.

Answer “yes” or “no” to the following.

1. Is 6 a factor of 30? 2. Is 4 a factor of 25?



Circle the numbers that are factors of the first number.

7. 8: 2 4 5 6 9 8. 20: 2 5 6 10 15

9. 17: 1 2 3 5 17 10. 40: 2 5 7 8 10

11. 32: 2 4 7 8 12 12. 48: 5 6 8 9 11

13. 25: 2 5 6 8 25 14. 60: 3 5 8 10 15

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Fractions & Mixed Numbers (cont.)


ACTIVITY 3 Divisibility

Tests for Divisibility: A number is divisible by 2 if the last digit is an even number. A number is divisible by 3 if the digits add up to any multiple of 3 (e.g., 3, 6, 9,…). A number is divisible by 5 if the last digit is 0 or 5. A number is divisible by 10 if the last digit is 0.

Circle the numbers in this row that are…

1. divisible by 2: 5 8 10 30 45 29 36 52 75 484

2. divisible by 3: 9 11 16 27 33 50 24 18 423 780

3. divisible by 5: 11 35 70 28 90 85 43 110 52 895

4. divisible by 10: 59 80 60 44 20 12 70 118 220 350

Bonus: Which of these numbers is divisible by 2 and 3? 28 36 63

ACTIVITY 4 Prime and Composite Numbers

A number is prime if it is a whole number greater than 1 that has only 1 and itself as factors. A number is composite if it is a whole number greater than 1 that has at least one other factor besides 1 and itself.

In this grid, circle all the prime numbers and underline all the composite numbers.

1 2 3 4 5 6 7

8 9 10 11 12 13 14

15 16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31 32 33 34 35

36 37 38 39 40 41 42

43 44 45 46 47 48 49

33

9

1211211

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ACTIVITY 5 Prime Factorization

Every composite number can be factored into a product of prime numbers. This is called prime factorization. To give the prime factorization of a com-posite number, first list the number as the product of any two numbers. Then, if either or both of those numbers is not prime, factor them. Continue until there are no more factors that are not prime.

Example: Find the prime factorization of 12: 12 = 2 x 6 (2 is prime, but 6 is not.)

Factor 6: 2 x 3 (Both are prime.)

Therefore, the prime factorization of 12 is 2 x 2 x 3.

Find the prime factorization for each number.

1. 44 2. 32 3. 45 4. 18 5. 50 6. 28

ACTIVITY 6 Using Factoring to Find the Greatest Common FactorThe Greatest Common Factor (GCF) of two numbers is the largest number that is a factor of both numbers. One way to find the Greatest Common Factor of two numbers is to list all the fac-tors of each number, and then find the factors they have in common and see which is the largest.

Example: Find the GCF of 12 and 18Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 18: 1, 2, 3, 6, 9, 18Factors in common: 1, 2, 3, 6 6 is the largest, so it is the GCF.

Use the method described to find the greatest common factor of these number pairs.

1. 20 and 36 2. 21 and 28

Factors of 20: Factors of 21:


Common factors: GCF: Common factors: GCF:

3. 16 and 42 4. 24 and 48



Common factors: GCF: Common factors: GCF:

Fractions & Mixed Numbers (cont.)Fractions & Mixed Numbers

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ACTIVITY 7 Using Prime Factorization to Find the Greatest Common FactorAnother way to find the Greatest Common Factor of two numbers is to first find the prime factorization of both numbers, list all the prime factors the numbers have in common, then multiply the common prime numbers together to get the GCF.

Use prime factorization to find the GCF of each of these number pairs.

1. 12 and 30 2. 21 and 35

Prime factors of 12: Prime factors of 21:


Prime factors in common: Prime factors in common:

Multiply common factors: Multiply common factors:

3. 18 and 24 4. 15 and 50



Prime factors in common: Prime factors in common:

Multiply common factors: Multiply common factors:

ACTIVITY 8 Least Common Multiple

The Least Common Multiple (LCM) of two or more numbers is the lowest number that each is a factor of (except 0). To find the Least Common

Multiple, begin listing the multiples of both numbers until you find the first one they have in common (the lowest multiple).

Example: 3 and 5 Multiples of 3: 3, 6, 9, 12, 15, 18, 21 Multiples of 5: 5, 10, 15, 20, 25 LCM: 15

Find the LCM of these pairs of numbers:

1. 4 and 6 2. 2 and 7

Multiples of 4: Multiples of 2:


LCM: LCM:

3. 3 and 8 4. 5 and 9



LCM: LCM:



GCF

LCM

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ACTIVITY 9 Equivalent Fractions

Complete to get an equivalent fraction.

1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

ACTIVITY 10 Simplifying Improper Fractions

Simplify these improper fractions.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

4 = 3 6

1 = 3 18

4 = 5 50

5 = 4 12

5 = 8 16

7 = 9 45

3 = 8 24

3 = 4 16

1 = 8 48

1 = 42

2 = 83

3 = 64

6 = 5

9 = 4

10 = 5

18 = 5

27 = 3

20 = 3

17 = 3

29 = 5

44 = 7

33 = 4

50 = 7

17 = 2



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ACTIVITY 11 Writing Whole Numbers as Fractions

To write a whole number as a fraction, write the number over the denominator 1, and then mul-tiply both the numerator and the denominator by the same number.

Example: Change 2 to fifths.

Change to halves.

1. 4 2. 6 3. 10

Change to the indicated number.

4. 6 to fifths 5. 8 to fourths 6. 3 to ninths

7. 7 to thirds 8. 4 to sixths 9. 2 to tenths

ACTIVITY 12 Writing Mixed Numbers as Fractions

To write a mixed number as a fraction, multiply the whole number by the denominator, and add the numerator. This number becomes the numerator and is placed over the original denominator.

Example: Change to a fraction.

Whole number (2) x denominator (4) = 8 + numerator (3) = 11

Then place 11 over the denominator (4).

Change each mixed number to a fraction.

1. 2. 3.

4. 5. 6.

7. 8. 9.

2 2 x 5 = 101 1 5 5

2 3 4

11 4

1 1 = 2

2 1 = 3

3 1 = 4

3 2 = 5

4 2 = 7

2 5 = 8

3 3 = 7

6 2 = 3

5 2 = 4



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ACTIVITY 13 Writing Fractions in Lowest Terms

To write a fraction in lowest terms, divide both the numerator and the denominator by their Greatest Common Factor.

Example:

Write each fraction in lowest terms.

1. 2. 3.

4. 5. 6.

7. 8. 9.

ACTIVITY 14 Adding Fractions With Like DenominatorsTo add fractions with like denominators, add the numerators, write the sum over the denomina-tor, and then reduce the fraction to lowest terms if necessary.

Example:

Add and put in simplest form.

1. 2. 3.

4. 5. 6.

7. 8. 9.

18 = 18 ÷ 6 = 324 24 ÷ 6 4

9 = 18

36 = 42

12 = 24

6 = 30

7 = 56

18 = 63

20 = 60

5 = 45

16 = 40

5 + 3 = 5 + 3 = 8 = 1 2 = 1 16 6 6 6 6 3

2 + 3 =8 8

4 + 1 =5 5

2 + 4 =9 9

3 + 3 =10 10

6 + 7 =12 12

5 + 2 =11 11

4 + 8 =15 15

13 + 7 =20 20

17 + 6 =18 18



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ACTIVITY 15 Adding Fractions With Unlike Denominators

To add fractions with unlike denominators, find the Least Common Multiple, change to equiva-lent fractions, and then add.

Example: = ? 12 and 6 have 12 as the LCM. stays the same.

Change to Add:

Add and put in simplest form.

1. 2. 3.

4. 5. 6.

7. 8. 9.

ACTIVITY 16 Subtracting Fractions With Like Denominators

To subtract fractions with like denominators, subtract the numerators, put the difference over the denominator, and reduce to lowest terms if necessary.

Example:

Subtract and put in simplest form.

1. 2. 3.

4. 5. 6.

7. 8. 9.

3 + 4 12 6

312

4 86 12

3 + 8 = 1112 12 12

1 + 1 =2 3

3 + 1 =8 4

2 + 1 =5 3

4 + 3 =5 8

2 + 2 =5 6

1 + 1 =8 16

5 + 2 =9 6

3 + 2 =12 4

3 + 5 =4 7

9 – 4 = 5 = 115 15 15 3

5 – 1 =8 8

4 – 2 =3 3

6 – 1 =8 8

9 – 4 =10 10

12 – 4 =20 20

8 – 6 =24 24

11 – 3 =32 32

15 – 4 =25 25

16 – 9 =28 28



5/10 - 3/10=2/10



Activity 1 (page 3) 1. yes 2. no 3. no 4. yes 5. no 6. yes 7. 2, 4 8. 2, 5, 10 9. 1, 17 10. 2, 5, 8, 1011. 2, 4, 8 12. 6, 8 13. 5, 25 14. 3, 5, 10, 15

Activity 2 (page 3) 1. 1, 2, 3, 4, 6, 8, 12, 24 2. 1, 2, 3, 5, 6, 10, 15, 30 3. 1, 2, 3, 6, 9, 18 4. 1, 5, 25 5. 1, 2, 3, 4, 6, 9, 12, 18, 36 6. 1, 3, 5, 9, 15, 45 7. 1, 2, 4, 5, 8, 10, 20, 40 8. 1, 29 9. 1, 3, 7, 21 10. 1, 2, 3, 4, 6, 12

Activity 3 (page 4)Divisible by 2: 8, 10, 30, 36, 52, 484Divisible by 3: 9, 27, 33, 24, 18, 423, 780Divisible by 5: 35, 70, 90, 85, 110, 895Divisible by 10: 80, 60, 20, 70, 220, 350Bonus: 36

Activity 4 (page 4)The following numbers should be circled: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. The rest of the numbers should be underlined.

Activity 5 (page 5) 1. 2 · 2 · 11 2. 2 · 2 · 2 · 2 · 2 3. 3 · 3 · 5 4. 2 · 3 · 3 5. 2 · 5 · 5 6. 2 · 2 · 7

Activity 6 (page 5) 1. Factors of 20: 1, 2, 4, 5, 10, 20 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 Common Factors: 1, 2, 4 GCF: 4 2. Factors of 21: 1, 3, 7, 21 Factors of 28: 1, 2, 4, 7, 14, 28 Common Factors: 1, 7 GCF: 7 3. Factors of 16; 1, 2, 4, 8, 16 Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42 Common Factors: 1, 2 GCF: 2 4. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 48: 1, 2, 3, 4, 6, 8, 12, 24, 48 Common Factors: 1, 2, 3, 4, 6, 8, 12, 24 GCF: 24

Activity 7 (page 6) 1. Prime factors of 12: 2, 3 Prime factors of 30: 2, 3, 5 Prime factors in common: 2, 3 GCF: 6 2. Prime factors of 21: 3, 7 Prime factors of 35: 5, 7 Prime factors in common: 7 GCF: 7 3. Prime factors of 18: 2, 3 Prime factors of 24: 2, 3 Prime factors in common: 2, 3 GCF: 6

4. Prime factors of 15: 3, 5 Prime factors of 50: 2, 5 Prime factors in common: 5 GCF: 5

Activity 8 (page 6) 1. Multiples of 4: 4, 8, 12, 16, 20, 24 Multiples of 6: 6, 12, 18, 24 LCM: 12 2. Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 Multiples of 7: 7, 14, 21 LCM: 14 3. Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27 Multiples of 8: 8, 16, 24, 32 LCM: 24 4. Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 Multiples of 9: 9, 18, 27, 36, 45, 54 LCM: 45

Activity 9 (page 7) 1. 8 2. 6 3. 12 4. 9 5. 15 6. 40 7. 10 8. 35 9. 6 10. 8 11. 12 12. 8

Activity 10 (page 7)

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7. 5@d 8. 5$g 9. 6@j 10. 8!f 11. 7!j 12. 8!s


1. *s 2. AwS 3. Sw: 4. Dt: 5. DrS

6. SoJ 7. SeA 8. SyF 9. WqPp


1. #s 2. &d 3. ArD 4. AtJ 5. Du:

6. SiA 7. SuF 8. Se: 9. SrS


1. !s 2. ^j 3. !s 4. !g 5. !k

6. @j 7. !d 8. !l 9. @g


1. %k 2. 1 3. #g 4. @d 5. 1 aQs

6. aUa 7. $g 8. 1 9. 1 aTk


1. %h 2. %k 3. QqQt 4. 1 rJp 5. QqQt

6. aEh 7. *l 8. #f 9. 1!s#k


1. !s 2. !s 3. @g 4. @d 5. %k

6. aQs 7. !f 8. !s!g 9. !f

Answer Keys

Answer Keys

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