Ch



N.RN.1

Selected Response

SELECT THE CORRECT ANSWER.

1. Write the radical expression in rational exponent form.

[pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

2. Write the radical expression in rational exponent form.

[pic]

[pic] [pic]

[pic] [pic]

[pic] k4

[pic] k10

3. Which values of p give the expression [pic] a real number result when simplified?

[pic] p ( 3

[pic] p ( 2

[pic] p ( 0

[pic] p ( 0

Select all correct answers.

4. Which of the following do not have integer exponents when rewritten in rational exponent form and simplified? Assume that s is nonnegative.

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

Match each radical expression with its equivalent rational exponent expression. Assume that w is nonnegative.

5. [pic]

6. [pic]

7. [pic]

8. [pic]

A [pic]

B [pic]

C [pic]

D [pic]

E [pic]

F [pic]

G [pic]

H [pic]

9. Given that the fourth root of x is defined as a quantity that, when raised to the fourth power, equals x, explain why it makes sense that [pic]

10. Let n ( 4m. Rewrite [pic] in rational exponent form and simplify. Assume that m is positive.

11. Given that the definition of the cube root of x is that it’s a quantity that, when raised to the third power, equals x, explain why it makes sense that [pic]

12. A student wrote the following:

[pic]

Based on this, the student claims that [pic]for all values of x.

a. Give an example of a value of x that makes this statement untrue and explain your reasoning.

b. Explain how you can restrict the value of x so that the student’s statement is true.

N.RN.2

Selected Response

SELECT THE CORRECT ANSWER.

1. Simplify [pic] Assume z is positive.

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

2. Which of the following is equal to

[pic] Assume that j is positive.

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

3. Write [pic] using rational exponents. Assume u and v are both positive.

[pic] [pic] [pic] [pic]

[pic] [pic] [pic] [pic]

Select all correct answers.

4. Which of the following are equal to

[pic] Assume that p is positive.

[pic] [pic] [pic] [pic]

[pic] [pic] [pic] [pic]

[pic] [pic] [pic] [pic]

Constructed Response

5. WRITE [pic] USING ONLY POSITIVE EXPONENTS. ASSUME C AND D ARE BOTH POSITIVE. SHOW ALL WORK.

6. Write the four expressions in descending order of resulting exponent when written in simplified rational exponent form. Assume t is positive.

[pic] [pic] [pic] [pic]

7. Which values of d give the expression [pic] a real number result when simplified? Explain your answer.

8. Show that [pic] for positive values of m, n, and a. Then use this information to simplify [pic] for positive values of j and k. Show all work.

9. On a recent exam, Terrell was asked to simplify [pic] assuming that x is not zero. His work is shown below.

[pic]

a. What mistake did Terrell make?

b. Find the correct answer. Show your work.

c. Are the original expression and the expression you found in part b equivalent when x is negative? Explain why or why not. (Hint: Check to see if both expressions have real number results with negative x.)

N.RN.3

Selected Response

SELECT THE CORRECT ANSWER.

1. Which of the following is not a rational number?

[pic] The product of 2 and [pic]

[pic] The sum of [pic] and [pic]

[pic] The sum of [pic] and [pic]

[pic] The product of 2 and [pic]

2. Which of the following is an irrational number?

[pic] The sum of 3 and 0.111....

[pic] The product of [pic] and width [pic]

[pic] The product of [pic] and [pic]

[pic] The sum of [pic] and [pic]

3. Which of the following shapes has an area that’s a rational number?

[pic] A triangle with base [pic] and

height (

[pic] A rectangle with length [pic] and width [pic]

[pic] A square with side length [pic]

[pic] A circle with diameter 8

Select all correct answers.

4. The perimeter of the triangle below is an irrational number.

[pic]

Which of the following are possible values of a and b?

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

Select the correct answer for each lettered part.

5. Determine whether each of the following are rational or irrational.

a. The product of[pic]and 5 [pic] Rational [pic] Irrational

b. [pic]evaluated at[pic] [pic] Rational [pic] Irrational

c. The sum of [pic] and[pic] [pic] Rational [pic] Irrational

d. [pic]evaluated at [pic] [pic] Rational [pic] Irrational

6. CLASSIFY[pic]AS RATIONAL OR IRRATIONAL. EXPLAIN YOUR REASONING.

7. Explain why the area of a circle with a rational radius must be an irrational number.

8. Given that the set of rational numbers is closed under addition, prove that the sum of a nonzero rational number and an irrational number is an irrational number.

9. Given that the set of rational numbers is closed under multiplication, prove that the product of a nonzero rational number and an irrational number is an irrational number.

N.Q.1*

Selected Response

SELECT THE CORRECT ANSWER.

1. A certain cooking oil has a density of

0.91 grams per milliliter. Which of the following series of calculations correctly determines the mass, in kilograms, of

15 liters of this oil?

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

2. Ordering n books from an online bookstore at $19.99 per book comes with a 6.25% sales tax and a shipping charge of $3.50 for each book after the first. If n is the whole number of books ordered, what are the units for the quantity represented by the expression

19.99n + 0.0625(19.99n) + 3.50(n − 1)?

[pic] dollars

[pic] books

[pic] dollars per book

[pic] percent of total cost

3. Shawn jogs n blocks, each of which are d meters long, in t minutes. What are the units for the expression [pic]Shawn’s average speed for his jog?

[pic] blocks

[pic] meters

[pic] minutes

[pic] meters per minute

Each activity will be graphed, with time on the horizontal axis. Match the activities with appropriate scales for the horizontal axis.

4. Graphing the remaining amount of a sandwich against the time taken to eat it

5. Graphing the distance traveled against the time taken to drive to work 30 miles away during rush hour

6. Graphing the distance traveled against the time taken to fly from the east coast of the United States to the west coast

7. Graphing the amount of weight lost against the time spent on a new diet and exercise regimen

8. Graphing the length of a signature against the time spent writing it

A 0 days to 60 days

B 0 seconds to 2 seconds

C 0 minutes to 10 minutes

D 0 hours to 24 hours

E 0 hours to 10 hours

F 0 minutes to 60 minutes

G 0 seconds to 30 seconds

H 0 days to 3 days

Select the correct answer for each lettered part.

9. A newton is a unit of force, and it’s measured in units of kilogram-meters per second squared, or [pic] Which of the following represent a quantity measured in newtons?

| a. [pic] | [pic] Yes [pic] No |

| b. [pic] | [pic] Yes [pic] No |

| c. [pic] | [pic] Yes [pic] No |

| d. [pic] | [pic] Yes [pic] No |

CONSTRUCTED RESPONSE

10. A local store sells muffins for $0.75 each. The graph below shows a customer’s total bill C as a function of m muffins purchased, which can be represented by the function C ’ 0.75m.

[pic]

Explain what the point at the origin represents.

11. Paige owns a car with a 12-gallon gas tank. Her car gets a highway gas mileage of 32 miles per gallon.

a. Write two ratios to represent the gas mileage of Paige’s car. Make sure to include the units for the quantities.

b. Paige wants to drive to her sister’s house, which is 162.4 highway miles away from where she lives. Use one of the ratios you wrote in part a to calculate how many gallons of gas Paige needs to make the trip.

c. Gas costs $3.50 per gallon. Write two ratios to represent the cost of a gallon of gas. Make sure to include units for the quantities.

d. Use one of your ratios from part c to calculate how much Paige’s trip costs. Round up to the nearest cent.

N.Q.2*

Selected Response

SELECT THE CORRECT ANSWER.

1. The math club is having a fundraiser, selling mugs for $5 each and T-shirts for $10 each. The club raised $1000. Which model describes the relationship between sales and money raised?

[pic] $5(the number of mugs sold) + $10(the number of T-shirts sold) ’ $15

[pic] $10(the number of mugs sold) + $5(the number of T-shirts sold) ’ $1000

[pic] $5(the number of mugs sold) + $10(the number of T-shirts sold) ’ $1000

[pic] $5(the number of mugs sold) − $10(the number of T-shirts sold) ’ $1000

2. Zach earns $10 for every lawn he rakes and $15 for every lawn he mows. He deposits $500 into his college fund at the end of the summer. Which model describes the relationship between work and money earned?

[pic] $15(lawns raked) + $10(lawns mowed) ’ $500

[pic] $10(lawns raked) + $15(lawns mowed) ’ $500

[pic] (lawns raked) + (lawns mowed) ’ $500

[pic] $10(lawns raked) + $15(lawns mowed) ’ $25

3. Susie’s Clothing Store sells sweatshirts for $30 and sweatpants for $25. The Drama Club buys a total of 100 sweatshirts and sweatpants and spends $2825. Which model describes this situation?

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

Match each situation with the correct expression.

Match each description of the growth of a culture of bacteria cells to an appropriate model. Assume the culture begins with a single cell.

| 4. The cell count doubles every hour. | A t + 2 |

| 5. The culture produces three more cells every hour. | B 3t |

| 6. The cell count triples every hour. | C (t + 1)2 |

| | D 2t |

| | E t + 3 |

| | F 3t + 1 |

Constructed Response

7. A FACTORY PRODUCES WIDGETS AND SPROCKETS. THE FACTORY SELLS WIDGETS FOR

$2 EACH, AND SPROCKETS FOR $3 EACH. THE TOTAL AMOUNT OF MONEY EARNED FROM SELLING WIDGETS AND SPROCKETS LAST MONTH WAS $3000.

a. Choose appropriate variables to represent the number of widgets sold and the number of sprockets sold. Write an equation representing the total amount of money earned from selling widgets and sprockets.

b. Write an appropriate model for the situation in which this factory sold 1225 widgets and sprockets to

earn $3000.

c. This factory also sells gizmos for $5. Choose an appropriate variable to represent the number of gizmos sold and write an equation representing the situation in which the total amount of money earned from

selling widgets, sprockets, and gizmos is $5000.

8. A model rocket’s height after being launched is modeled by a quadratic function.

a. One quantity of interest is maximum height. What other quantity might be of interest?

b. The maximum height of the rocket’s flight path is 192 meters after

8 seconds. Find and choose the dependent and independent variables that represent the quantities in this problem.

Include units.

c. Write a function that models the height of the rocket as it relates to the time since the rocket was launched, given that the rocket starts at a height of 0. Write the function in the form [pic]

d. This rocket is being fired off in a certain town. In the interest of safety, town regulations dictate that model rockets should land no more than

20 meters from where they are launched. This rocket’s horizontal speed during flight is 1.5 meters per second. Does this rocket meet local regulations? Explain your reasoning.

N.Q.3*

Selected Response

SELECT THE CORRECT ANSWER.

1. A triangle has side lengths 2.02 cm,

3.570 cm, and 4.1 cm. What is the perimeter of this triangle to the correct number of significant digits?

[pic] 9.69

[pic] 9.690

[pic] 9.7

[pic] 10

2. Which of these measurements is the most precise?

[pic] 4 m

[pic] 127 mm

[pic] 1.3 km

[pic] 5.14 cm

3. A rectangle has a length of 4.2 feet and a width of 7.36 feet. How many significant digits does the area of the rectangle have?

[pic] 2 [pic] 5

[pic] 3 [pic] 6

Select all correct answers.

4. Which of the following calculated values will have three significant digits?

[pic] The perimeter of a square with side length 1.02 ft

[pic] The area of a square with side length 0.024 m

[pic] The perimeter of a triangle with side lengths 84.5 cm, 94 cm, and 117 cm

[pic] The area of a triangle with base 4.50 in. and height 10.02 in.

[pic] The circumference of a circle with radius 0.0910 m

[pic] The area of a circle with radius

5000 ft

Select the correct answer for each lettered part.

5. Classify each measurement as having more significant digits than, fewer significant digits than, or the same number of significant digits as the volume of a cube with side length 14.20 ft.

| a. The surface area of a sphere with radius 4.2 m | [pic] More |

| |[pic] Fewer |

| |[pic] Same |

| b. The sum of dimensions 2.049 ft and 10.67 ft | [pic] More |

| |[pic] Fewer |

| |[pic] Same |

| c. Half of the length 175.08 m | [pic] More |

| |[pic] Fewer |

| |[pic] Same |

| d. The sum of lengths 12.125 mm and 10 mm | [pic] More |

| |[pic] Fewer |

| |[pic] Same |

| e. The product of 120.7 cm, 44.50 cm, and 1.553 cm | [pic] More |

| |[pic] Fewer |

| |[pic] Same |

constructed Response

6. DO 38,000 CM, 38 CM, 0.038 CM, AND 0.00038 CM ALL HAVE THE SAME NUMBER OF SIGNIFICANT DIGITS? EXPLAIN YOUR REASONING, INCLUDING THE NUMBER OF SIGNIFICANT DIGITS IN EACH MEASUREMENT.

7. A company produces parts for an automobile manufacturer. One part consists of two rods connected end to end with a joint. The lengths for the rods are 31.4 cm and 82.25 cm.

a. What is the combined length of the two rods, using the correct number of significant digits? (Assume that the joint doesn’t add any additional length.) Show your work.

b. The automobile company says that the combined length of the two rods must be within 0.01 cm of 113.65 cm. The manager of the company says that this level of precision isn’t possible given the precision of the lengths of the individual parts. Is the manager correct? Explain why or why not.

8. Two physics classes at two different schools have a competition to build the best catapult. Each class will throw rubber balls with the same weight, and both will use a tape measure marked in eighths of an inch to measure the results.

At his school, Randal fires his catapult and measures the distance it throws the ball. The edge of the ball falls between the marks for 13 feet [pic] inches and

13 feet [pic] inches, so he rounds to the eighth of an inch closest to where his ball landed, which is 13 feet [pic] inches. At her school, Lacey fires her catapult and measures the distance. Her distance falls between the same two marks, but she reports her distance as 13 feet

[pic] inches and claims she’s won.

a. What measurement error did

Lacey make?

b. The judges declare a tie, but is it possible that Randal would have won if they used a measuring tape with higher precision? Explain why or

why not.

N.RN.1 Answers

1. B

2. A

3. D

4. C, E, F

5. A

6. B

7. G

8. D

9. [pic]so [pic] is the fourth root of b by definition. Thus, it makes sense that [pic]

Rubric

3 points for a logically sound explanation

10.

[pic]

Rubric

1 point for rewriting [pic] as [pic]

1 point for rewriting [pic] as [pic]

1 point for rewriting [pic] as [pic](Score similarly if students choose to substitute [pic] for n and then rewrite and simplify.)

11. [pic]which means that [pic] is the cube root of [pic] by definition. Thus, it makes sense that [pic]

Rubric

3 points for a logically sound explanation

12. a. Use any negative number for x, for example:

[pic]

b. The statement is true only for nonnegative values of x.

Rubric

a. 1 point for giving an example of a

number that makes the statement true;

1 point for showing why the given

value doesn’t work

b. 2 points for a correct restriction

N.RN.2 Answers

1. B

2. D

3. A

4. A, E

5.

[pic]

Rubric

1 point for the correct answer; 2 points for showing appropriate work

6. [pic]

Rubric

2 points for the correct answer

7. The expression [pic] has a real number result when d ( 0.

If d is negative, then [pic] is also negative. If [pic] is negative, then [pic] is also negative. Raising a quantity to [pic] is the same as taking the fourth root, and there is no real fourth root of a negative number.

Rubric

1 point for the correct answer; 2 points for a logically sound explanation

8.

[pic]

[pic]

Rubric

2 points for showing [pic]

2 points for showing [pic]

9. a. In the first step, Terrell divided the exponents when he should have subtracted them.

b.

[pic]

c. They are equivalent; in the expression [pic] a negative value of x means the numerator will be negative and the denominator will be positive (because

it is the equivalent of taking the 5th root of the square of a negative number, which is always positive), giving a negative result. In the expression [pic]a negative value of x simplifies to 1 over a negative number, which also gives a negative result.

Rubric

a. 1 point for identifying the error

b. 1 point for answer; 1 point for adequate work

c. 1 point for answer; 2 points for appropriate explanation

N.RN.3 Answers

1. D

2. D

3. C

4. C, D

5. a. Irrational

b. Rational

c. Irrational

d. Irrational

6. The expression[pic]is rational.

[pic]

46 is a rational number.

Rubric

1 point for identifying the expression as rational; 2 points for simplifying [pic]to 46 and noting that 46 is rational

7. Since the radius of the circle is rational, it can be written in the form [pic] where a and b are nonzero integers. Then the square of the radius of the circle can be written in the form [pic] which is rational since the set of rational numbers is closed under multiplication. The area of the circle can then be written in the form [pic] Since the area of the circle is the product of a nonzero rational number and an irrational number, it must be irrational.

Rubric

3 points for a logically correct argument

8. Let a be rational, let b be irrational, and let a ( b ( c. Assume c is rational. Rewrite a ( b ( c as b ( c ( (a). Since a is rational, −a is rational, and since the set of rational numbers is closed under addition, c ( (−a) is rational. Thus, b is rational. However, this contradicts the condition that b is irrational. Thus, the assumption that c is rational must be invalid. This means that c, the sum of a rational number and an irrational number, must be irrational.

Rubric

5 points for a logically correct argument

9. Let a ( 0 be rational, let b be irrational, and let ab ( c. Assume c is rational. Rewrite ab ( c as [pic] Since a is rational and not zero,[pic] is rational, and since the set of rational numbers is closed under multiplication,[pic] is rational. Thus, b is rational. However, this contradicts the assumption that b is irrational. Thus, the assumption that c is rational must be invalid. This means that c, the product of a nonzero rational number and an irrational number, must be irrational.

Rubric

5 points for a logically correct proof

N.Q.1* Answers

1. D

2. A

3. D

4. C

5. F

6. E

7. A

8. B

9. a. Yes

b. No

c. Yes

d. Yes

10. The point on the origin represents purchasing 0 muffins for $0.00.

Rubric

1 point each for interpreting what zero means on both the x and y axes

11. a. [pic][pic]

b. [pic]

c. [pic][pic]

d. [pic]

Rubric

1 point for each part

N.Q.2* Answers

1. C

2. B

3. A

4. D

5. F

6. B

7. a. Let w be the number of widgets sold and s be the number of sprockets sold. An expression for the amount of money earned from selling w widgets is 2w. An expression for the amount of money earned from selling s sprockets is 3s. The sum of these two amounts is $3000. Therefore, 2w + 3s ’ 3000.

b. [pic]

c. Let g be the number of gizmos sold. An expression for the amount of money earned from selling g gizmos is 5g. The sum of this amount and the two amounts from part a is $5000. Therefore, 2w + 3s + 5g ’ 5000.

Rubric

a. 1 point for the variables;

1 point for the equation

b. 1 point for each equation

c. 1 point for the variable;

1 point for the equation

8. a. duration of flight

b. Since the distance that the rocket travels is dependent on time, the independent variable is time in seconds and is denoted by the variable t. The dependent variable is the vertical distance that the rocket travels in meters and is denoted by the

variable h.

c. Since the maximum height of the rocket is 192 meters after 8 seconds, the vertex of the parabola that represents this relationship is (8, 192). Substitute 8 and 192 into the general form for a parabola, f(x) ’ a(x − h)2 + k. h(t) ’ a(t − 8)2 + 192

To find the value of a, substitute 0 for t and 0 for h(t) and solve for a.

[pic]

Since a ’ −3, then the equation that models this situation is

h(t) ’ −3(t − 8)2 + 192.

d. The rocket takes off and lands when the value of h(t) is zero.

[pic]

The rocket is being fired at t ’ 0, so it lands at t ’ 16 seconds. The rocket’s horizontal speed is 1.5 meters per second, so it travels 1.5 ( 16 ’

24 meters horizontally in that time. The rocket does not meet local regulations.

Rubric

a. 1 point

b. 1 point for variables

c. 2 points for equation

d. 1 point for correct conclusion;

2 points for explanation, including the calculation of the distance traveled

N.Q.3* Answers

1. C

2. D

3. A

4. A, C, D, E

5. a. Fewer

b. Same

c. More

d. Fewer

e. Same

6. Each measurement has only two significant digits, 3 and 8, so they all have the same number of significant digits. The zeros in 38,000 cm are not significant because zeros at the end of whole numbers are generally not considered significant. The zeros in 0.038 cm and 0.00038 cm are not significant because they are not to the right of the first nonzero digit after the decimal point.

Rubric

1 point for a “Yes” conclusion to the question; 0.5 point for finding the correct number of significant digits in each measurement; 0.5 point for giving correct reasoning for the number of significant digits in each measurement

7. a. When adding measurements, round the sum to the last significant digit of the least precise measurement. The least precise measurement is 31.4 cm, so round to the tenths place.

[pic]

b. The manager is correct. The precision requirement is to the nearest hundredth of a centimeter, but the precision for combining the two parts can only be determined to the nearest tenth of a centimeter.

Rubric

a. 1 point for correct sum; 1 point for appropriate work

b. 1 point for noting that the manager is correct; 2 points for an appropriate explanation

8. a. Lacey reported her distance to a precision that her measuring tool does not possess. She cannot determine the distance of the throw in sixteenths of an inch with a tape measure that is only marked in eighths of an inch.

b. Yes; Randal rounded his measurement to the nearest eighth of an inch, but this was only an estimate based on his measuring tool. It’s possible that Randal’s measurement would have been 13 feet [pic] inches when measured to the nearest sixteenth of an inch, and it’s also possible that Lacey overestimated her result and would have had a distance that rounded to 13 feet [pic] inches ([pic]inches). In this case, Randal would have won.

Rubric

a. 2 points for identifying and explaining the error

b. 1 point for noting that Randal could win, and 3 points for giving a plausible situation that supports that conclusion

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