Download Lincoln Public Schools Math 8 McDougall Littell Middle School Math

Lincoln Public Schools Math 8 McDougall Littell Middle School Math Course 3 – Chapter 4 Items marked A, B, C are increasing in difficulty. Group “A” questions are the most basic while Group “C” are the most difficult and require higher levels of thinking skills. The level of difficulty is only relative to the same section. All problems include random number generation. [n] indicates n problems types are available in the topic. Note: Some problems use text boxes for fractions and scientific notation. For these problems, do not enter spaces before or after the numeric values. Section 4.1 Objective: Students will be able to write the prime factorization of a number. 4_1 Prime Factors (3) – [10] Finding the prime factors of a number with three prime factors. 4_1 Prime Factors (4) – [10] Finding the prime factors of a number with four prime factors. Section 4.2 Objective: Students will be able to find the greatest common factor. 4_2 Greatest Common Factor (4) – [10] Find the GCF when each factor has four prime factors. 4_2 Greatest CF with variables – [4] Find the GCF of an expression only involving variables. 4_2 GCF with numbers and 1 variable – [4] Find the GCF of an expression with a number and one variable. 4_2 GCF with numbers and variables – [4] Find the GCF of an expression with a number and multiple variables. Section 4.3 Objective: Students will be able to simplify fractions. 4_3 Simplify Fractions – [10] Simplify fractions by finding the GCF. Section 4.4 Objective: Students will be able to find the least common multiple. 4_4 Find LCM of 2 numbers – [4] Finding the LCM of 2 numbers. 4_4 Find LCM of variables – [4] Finding the LCM of two variable expressions. 4_4 Find LCM of numbers and variables – [4] Finding the LCM of number and variable expressions. Section 4.5 Objective: Students will be able to compare and order fractions and mixed numbers. 4_5 Review Mixed to Improper Fraction – [3] Review converting mixed numbers to improper fractions. 4_5 Compare Fractions – [3] Use inequalities to compare fractions. 4_5 Compare Mixed Fractions – [3] Use inequalities to compare mixed fractions. 4_5 Order Fractions – [3] Order fractions from smallest to largest.
Section 4.6 Objective: Students will be able to multiply and divide expressions with exponents. 4_6 Multiply with Exponents (A) – [2] Multiply variable only expressions. 4_6 Multiply with Exponents (B) – [2] Multiply expressions with numbers and variables. 4_6 Multiply with Exponents (C) – [2] Multiply expressions with numbers and multiple variables. 4_6 Divide with Exponents (A) – [2] Divide variable only expressions. 4_6 Divide with Exponents (B) – [2] Divide expressions with numbers and variables. 4_6 Divide with Exponents (C) – [2] Divide expressions with numbers and multiple variables. Section 4.7 Objective: Students will be able to simplify expressions with negative exponents. 4_7 Negative Exponents Eval – [2] Evaluate single variables with negative exponents. 4_7 Mult Neg Exponents Vars – [2] Evaluate multiple variables with negative exponents. 4_7 Div Neg Exponents Vars – [2] Dividing with negative exponents. Section 4.8 Objective: Students will be able to read and write numbers using scientific notation. 4_8 Scientific Notation – [6] Writing numbers in scientific notation. 4_8 Scientific Notation Calculations – [2] Performing calculations in scientific notation.
Lincoln Public Schools – Math 8 – McDougal Littell Middle School Math Course 3 Please note: This demo is a one problem sample from each topic. All problems are random number problems and consist of multiple types for each topic. Some fraction problems are not properly formatted in this demo due to the conversion to Word form. They will appear properly formatted when used in EDU. 4_1 Prime Factors (3) – [10] Finding the prime factors of a number with three prime factors. Write the prime factors of 75. Write the value separated by a semi­colon ; as 3; 5; 7 Your Answer: 3; 5; 5 Correct Answer: 3; 5; 5 Comment: Write the prime factors of 75. Write the value separated by a semi­colon ; as 3; 5; 7 75 = 25 • 3 = 5 • 5 • 3 Enter the prime factors as 3; 5; 5 4_1 Prime Factors (4) – [10] Finding the prime factors of a number with four prime factors. Write the prime factors of 60. Write the value separated by a semi­colon ; as 3; 5; 7 Your Answer: 2; 2; 3; 5 Correct Answer: 2; 2; 3; 5 Comment: Write the prime factors of 60. Write the value separated by a semi­colon ; as 3; 5; 7 60 = 6 • 10 = (2 • 3) • (2 • 5) = 2 • 3 • 2 • 5 Enter the prime factors as 2; 2; 3; 5
4_2 Greatest Common Factor (4) – [10] Find the GCF when each factor has four prime factors. Find the greatest common factor of 30 and 42. Your Answer: 6 Correct Answer: 6 Comment: Find the greatest common factor of 30 and 42. Find the prime factors of each number. 30 = 2 • 3 • 5 42 = 2 • 3 • 7 Multiply the common prime factors. G.C.F. = 2 • 3 = 6 4_2 Greatest CF with variables – [4] Find the GCF of an expression only involving variables. Find the greatest common factor of 198 and 330. Your Answer: 66 Correct Answer: 66 Comment: Find the greatest common factor of 198 and 330. Find the prime factors of each number. 198 = 2 • 3 • 3 • 11 330 = 2 • 3 • 5 • 11 Multiply the common prime factors. G.C.F. = 2 • 3 • 11 = 66 4_2 GCF with numbers and 1 variable – [4] Find the GCF of an expression with a number and one variable. Find the greatest common factor of x 9 y 8 z 4 and x 3 y 4 z 10 . Enter the exponents using the ^ key. For example a 5 b 6 would be entered as a^5 b^6. Your Answer: x^3 y^4 z^4 Correct x^3 y^4 z^4 Answer: Comment: Find the greatest common factor of x 9 y 8 z 4 and x 3 y 4 z 10 . Enter the exponents using the ^ key. For example a 5 b 6 would be entered as a^5 b^6. The GCF is the lowest power of each variable. If the variable is not present, the exponent is 0 and is not included in the GCF The lowest power of x is 3, the lowest power of y is 4 and the lowest power of z is 4. The LCM is x 3 y 4 z 4 entered as x^3 y^4 z^4. 4_2 GCF with numbers and variables – [4] Find the GCF of an expression with a number and multiple variables.
Find the greatest common factor of 8x 2 and 20x 2 . Enter the exponents using the ^ key. For example a 5 b 6 would be entered as a^5 b^6. Your Answer: 4x^2 Correct Answer: 4 x^2 Comment: Find the greatest common factor of 8x 2 and 20x 2 . Enter the exponents using the ^ key. For example a 5 b 6 would be entered as a^5 b^6. The GCF of 8 and 20 is 4. The GCF is the lowest power of the variable. The lowest power of x is 2. The LCM is 4x 2 entered as 4 x^2. 4_2 GCF with numbers and variables – [4] Find the GCF of an expression with a number and multiple variables. Find the greatest common factor of 8x 3 y 10 and 20x 2 y 3 . Enter the exponents using the ^ key. For example a 5 b 6 would be entered as a^5 b^6. Your Answer: 4x^2 y^3 Correct 4 x^2 y^3 Answer: Comment: Find the greatest common factor of 8x 3 y 10 and 20x 2 y 3 . Enter the exponents using the ^ key. For example a 5 b 6 would be entered as a^5 b^6. The GCF of 8 and 20 is 4. The GCF is the lowest power of each variable. If the variable is not present, the exponent is 0 and is not included in the GCF The lowest power of x is 2 and the lowest power of y is 3. The LCM is 4x 2 y 3 entered as 4 x^2 y^3.
4_3 Simplify Fractions – [10] Simplify fractions by finding the GCF. Your answer: Your response Correct response 35 35 Simplify the fraction: Simplify the fraction: 40 40 Enter a fraction such as 2/3 in the textbox. Do not use spaces in the answer! Enter a fraction such as 2/3 in the textbox. Do not use spaces in the answer! 7/8 7/8 Comment: 35 Simplify the fraction: 40 Enter a fraction such as 2/3 in the textbox. Find the GCF of 35 and 40. The GCF is 5. Divide numerator and denominator by 5. The simplified fraction is 7/8. 4_4 Find LCM of 2 numbers – [4] Finding the LCM of 2 numbers. Find the least common multiple of 4 and 12. Your Answer: 12 Correct Answer: 12 Comment: Find the least common multiple of 4 and 12. Write the multiples of each number. 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48,.... 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, .... Choose the first number which is displayed in both lists. The LCM is 12.
4_4 Find LCM of variables – [4] Finding the LCM of two variable expressions. Find the lowest common multiple of x 3 y 8 and x 8 y 5 z 2 Enter the exponents using the ^ key. For example a 5 b 6 would be entered as a^5 b^6. Your Answer: x^8 y^8 z^2 Correct x^8 y^8 z^2 Answer: Comment: Find the lowest common multiple of x 3 y 8 and x 8 y 5 z 2 Enter the exponents using the ^ key. For example a 5 b 6 would be entered as a^5 b^6. The LCM is the highest power of each variable. The highest power of x is 8 and the highest power of y is 8. The highest power of z is 2. The LCM is x 8 y 8 z 2 entered as x^8 y^8 z^2. 4_4 Find LCM of numbers and variables – [4] Finding the LCM of number and variable expressions. Find the least common multiple of 6x 4 y 7 and 21x 3 y 6 . Enter the exponents using the ^ key. For example a 5 b 6 would be entered as a^5 b^6. Your Answer: 42x^4 y^7 Correct Answer: 42 x^4 y^7 Comment: Find the least common multiple of 6x 4 y 7 and 21x 3 y 6 . Enter the exponents using the ^ key. For example a 5 b 6 would be entered as a^5 b^6. The LCM of 6 and 21 is 42. The LCM of the variables is the highest exponent of each x 4 y 7 Putting the these together, the LCM is 42x 4 y 7 entered as 42 x^4 y^7
4_5 Review Mixed to Improper Fraction – [3] Review converting mixed numbers to improper fractions. 3 Write 3 as an improper fraction. 5 Enter a fraction into the answer box as 3/5. Your Answer: 18/5 Correct Answer: 18/5 Comment: 3 Write 3 as an improper fraction. 5 Enter a fraction into the answer box as 3/5. 3 • 5 + 3 = 18 18 The fraction is entered as 18/5.
5 4_5 Compare Fractions – [3] Use inequalities to compare fractions. Replace ? with <, =, or >. 2 1 ? 11 7 Your Answer: > Correct Answer: > Replace ? with <, =, or >. Comment: 2 1 ? 11 7 The least common denominator is 77. Multiply each fraction to get the least common denominator. 2•7 1•11 ? 11•7 7•11 14
11
? 77 77 14
Since 14 > 11, 11
> 77 2 and we can say 77 1 > 11 7
4_5 Compare Mixed Fractions – [3] Use inequalities to compare mixed fractions. Replace ? with <, =, or >. 2 4 120 ? 7 28 Your Answer: = Correct Answer: = Replace ? with <, =, or >. Comment: 2 120 4 ? 7 28 Convert to a mixed number. 30 120 ? 7 28 The least common denominator is 28. Multiply each fraction to get the least common denominator. 30•4 120•1 ? 7•4 28•1 120 120 ? 28 28 120 Since 120 = 120, 120 = 28 30 and we can say 28 120 = 7 28
4_5 Order Fractions – [3] Order fractions from smallest to largest. Your response Correct response Order the fractions from smallest to largest: Order the fractions from smallest to largest: 1 1 A. 2 A. 2 3 3 13 13 B. B. 5 5 12 12 C. C. 5 5 Smallest Fraction: A (33%) Middle Fraction: C (33%) Largest Fraction: B (33%) Smallest Fraction: A Middle Fraction: C Largest Fraction: B Comment: Order the fractions from smallest to largest. 1 2 13 , 3 12 , 5 5 The least common multiple is 15. Writing each with the LCM as a common denominator: 1 2 7 = 3 = 3 13 5 39
= 5 • 3 = 12 • 3 15 = 35
= 3 • 5 13 • 3 = 12 7 • 5 36
15 5 5 • 3 15 1 From smallest to largest, the fractions are: 2 12 , 3 13 , 5 5 4_6 Multiply with Exponents (A) – [2] Multiply variable only expressions. Simplify the expression: x 5 • x 2 Write expression with an exponent. Enter x 3 as x^3. Your Answer: x^7 Correct Answer: x^7 Comment: Simplify the expression: x 5 • x 2 Write expression with an exponent. Enter x 3 as x^3. When multiplying with exponents, add the exponents. x 5 • x 2 = x (5 + 2) = x 7 entered as x^7 4_6 Multiply with Exponents (B) – [2] Multiply expressions with numbers and variables. Simplify the expression: 2 4 y 6 • 2 4 y 3 Write expression with an exponent. Enter x 3 as x^3. Your Answer: 256y^9 Correct Answer: 256y^9 Comment: Simplify the expression: 2 4 y 6 • 2 4 y 3 Write expression with an exponent. Enter x 3 as x^3. When multiplying with exponents, add the exponents. 2 4 y 6 • 2 4 y 3 = 2 (4 + 4) • y (6 + 3) = 2 8 • y 9 = 256y 9 entered as 256y^9
4_6 Multiply with Exponents (C) – [2] Multiply expressions with numbers and multiple variables. Simplify the expression: 4 4 w 5 x 3 • 4 3 w 2 x 3 Write expression with an exponent. Enter x 3 as x^3. Your Answer: 16384w^7x^6 Correct Answer: 16,384 w^7 x^6 Comment: Simplify the expression: 4 4 w 5 x 3 • 4 3 w 2 x 3 Write expression with an exponent. Enter x 3 as x^3. When multiplying with exponents, add the exponents. 4 4 w 5 x 3 • 4 3 w 2 x 3 = 4 (4 + 3) • w (5 + 2) • x (3 + 3) = 4 7 • w 7 • x 6 = 16,384w 7 x 6 entered as 16,384 w^7 x^6 4_6 Divide with Exponents (A) – [2] Divide variable only expressions. Simplify the expression: x 7 x 4 Write expression with an exponent. Enter x 3 as x^3 Your Answer: x^3 Correct Answer: x^3 Comment: Simplify the expression: x 7 x 4 Write expression with an exponent. Enter x 3 as x^3 When dividing with exponents, subtract the exponents. x 7 = 4 x x (7­4) = x 3 entered as x^3
4_6 Divide with Exponents (B) – [2] Divide expressions with numbers and variables. Simplify the expression: 3 6 y 4 3 4 y 2 Write expression with an exponent. Enter x 3 as x^3 Your Answer: 9y^2 Correct Answer: 9y^2 Comment: Simplify the expression: 3 6 y 4 3 4 y 2 Write expression with an exponent. Enter x 3 as x^3 When dividing with exponents, subtract the exponents. 3 6 y 4 = 4 2 3 y 3 (6 ­ 4) • y (4 ­ 2) = 3 2 • y (4 ­ 2) = 9y 2 entered as 9y^2
4_6 Divide with Exponents (C) – [2] Divide expressions with numbers and multiple variables. Simplify the expression: 5 7 x 5 y 7 5 2 x 3 y 5 Write expression with an exponent. Enter x 3 as x^3 Your Answer: 3125x^2y^2 Correct Answer: 3,125x^2 y^2 Comment: Simplify the expression: 5 7 x 5 y 7 5 2 x 3 y 5 Write expression with an exponent. Enter x 3 as x^3 When dividing with exponents, subtract the exponents. 5 7 x 5 y 7 = 2 3 5 5 x y 5 (7 ­ 2) • x (5 ­ 3) • y (7 ­ 5) = 5 5 • x (5 ­ 3) • y (7 ­ 5) = 3,125x 2 y 2 entered as 3,125x^2 y^2 4_7 Negative Exponents Eval – [2] Evaluate single variables with negative exponents. Evaluate: 2 8 • 2 ­5 Enter fractions as 1/18 or 1/x^2. Your Answer: 8 Correct Answer: 8 Comment: Evaluate: 2 8 • 2 ­5 Enter fractions as 1/18 or 1/x^2. When multiplying with exponents, add the exponents. 2 8 • 2 ­5 = 2 (8+(­5)) = 2 3 = 8 4_7 Mult Neg Exponents Vars – [2] Evaluate multiple variables with negative exponents.
Write with positive exponents : x 2 • x ­8 Enter fractions as 1/18 or 1/x^2. Your Answer: 1/x^6 Correct Answer: 1/x^6 Comment: Write with positive exponents : x 2 • x ­8 Enter fractions as 1/18 or 1/x^2. When multiplying with exponents, add the exponents. x 2 • x ­8 = x (2+ (­8)) = x ­6 = When a negative exponent results, recall that 1 ­n a = a n 1 ­6 x = x 6 Enter the answer as 1/x^6 4_7 Div Neg Exponents Vars – [2] Dividing with negative exponents. Simplify the expression: a 6 a ­2 Write expression with an exponent. Enter x 3 as x^3 Your Answer: a^8 Correct Answer: a^8 Comment: Simplify the expression: a 6 a ­2 Write expression with an exponent. Enter x 3 as x^3 When dividing with exponents, subtract the exponents.
a 6 = ­2 a a (6­(­2)) = a 8 entered as a^8 4_8 Scientific Notation – [6] Writing numbers in scientific notation. Your answer: Your response Correct response Write 0.00000251 in scientific notation as 2.51 X 10 ­6 . Do not enter spaces in the answer boxes. Write 0.00000251 in scientific notation as 2.51 X 10 ­6 . Do not enter spaces in the answer boxes. Comment: Write 0.00000251 in scientific notation as 2.51 X 10 ­6 . Do not enter spaces in the answer boxes. The decimal place must be moved 6 places to the right. Therefore, the decimal is 2.51 and the power of 10 is ­6. 4_8 Scientific Notation Calculations – [2] Performing calculations in scientific notation. Your answer: Your response Correct response Write the answer in scientific notation: Write the answer in scientific notation: (­7.41 X 10 3 ) ÷ (1.9 X 10 ­7 ) (­7.41 X 10 3 ) ÷ (1.9 X 10 ­7 ) Do not use spaces in the answer boxes. Do not use spaces in the answer boxes. ­3.9 X 10 10 ­3.9 X 10 10 Comment: Write the answer in scientific notation: (­7.41 X 10 3 ) ÷ (1.9 X 10 ­7 ) Do not use spaces in the answer boxes. Divide the decimals and subtract the exponents since we are dividing. (­7.41) ÷ (1.9) = ­3.9 (3) ­ (­7) = 10 The answer is ­3.9 X 10 10