Rational and Irrational Numbers Math B College Prep

Module #6 Homework 2015-2016

Created in collaboration with Utah Middle School Math Project A University of Utah Partnership Project

San Dieguito Union High School District

6.1A Homework: Square Areas* Name:

Period:

1. Find the areas of the following shapes. On the grid, a horizontal or vertical segment joining two dots has a length of 1. Put your answers on the lines provided below the grid

A D

C B

A:

B:

C:

D:

2. List the first 12 perfect square numbers. 3. What is the side length of a square with an area of 8100 units2? 4. What is the area of a square with a side length of 0.06 units? 5. What is the side length of a square with area 35? 6. Show two different methods for finding the area of the shape below. Method #1

Method #2

2

7. Explain in words how finding the square root of a number is related to squaring a number.

Directions: Simplify the following. Assume all variables are positive. 8. √

9. √

10.

√

11. √

12. √

13. √

14. √

15. √

16. √

17. √

18. √

19. √

20. A checkerboard is a square made up of 32 black and 32 red squares. Assume that each square has a side length of 1 unit. a. What is the total area of the checkerboard?

b. What is the side length of the checkerboard?

c. Explain how your answers to part

and

help you determine the square root of 64.

3

Spiral Review: 21. Solve:

22. Write a linear equation in Standard Form that represents the table of values below: x

0

100

200

300

y

1.5

36.5

71.5

106.5

23. Complete the equation so that it has no solution: (

)

24. Write the equation of the line in slope-intercept form that passes through the points (2, 7) and (6, 15).

25. Sam spent $5.26 on some apples priced at $0.64 each and some oranges priced at $0.45 each. At another store, he could have bought the same number of apples at $0.32 each and the same number of oranges at $0.39 each for a total of $3.62. How many apples and oranges did Sam buy?

4

6.1B Homework: The Rational Number System* Name:

Period:

Directions: Determine if the given statement is true or false. If false, explain why. 1. All whole numbers are natural numbers.

2. Some rational numbers are integers.

3. All repeating decimals can be written as a fraction.

4.

is a rational number.

Directions: Change the following rational numbers into decimals without the use of a calculator. Show all of your work. 5.

6. 6.

8.

9.

7.

10.

5

Directions: Change the following decimals into fractions without the use of a calculator. Show all of your work. 11.

0.02

12.

13.

̅̅̅̅

14.

15.

̅

16.

̅

18.

̅̅̅̅

17.

̅

Directions: Simplify. Express answers as simplified fractions. 19.

̅

20.

̅

21.

̅

6

Spiral Review: 22. A line passes through the points (6, 3), (-4, -2) and (n, 1). What is the value of n?

23. Is the table of values proportional? Explain why or why not.

x y

2 7

4 11

6 15

24. Solve:

25. Both the points (2,-2) and (5,-4) are solutions of a system of linear equations. What conclusions can you make about the equations and their graphs.

7

6.1C Homework: Expanding Our Number System* Name:

Period:

Directions: Classify the following numbers as rational or irrational and provide a justification. Number

Natural number

Whole number

Integer

Rational number

Irrational number

Real

Justification

1. √

2. √

3.

4.

5. √

6.

7.

√

̅

8.

9. The number half-way between 0 and -1

10. The number that represents 7 degrees below 0. 11. The side length of a square with an area of 24

8

Directions: Represent the given set on a number line. 12. natural numbers less than 5

13. real numbers from -2 to 4

14. whole numbers from -3 to 3

Directions: Determine if the given statement is true or false. If false, re-write the statement to make it true. 15. The number

is rational.

16. A calculator can be used to determine whether a number is rational or irrational by looking at its decimal expansion.

17. The number

̅ is irrational because its decimal expansion goes on forever.

18. The number half-way between 3 and 4 is rational.

Directions: Use your knowledge of the Real Number System to answer the following questions 19. When can a number be a whole number, integer and a rational number? Explain your reasoning.

20. When can a number only be an integer and a rational number? Explain your reasoning.

21. When can a decimal be an irrational number? Explain your reasoning.

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22. When can a number be only a rational number? Explain your reasoning.

23. Can a number be a rational number and an irrational number? Explain your reasoning.

Directions: Create four statements about rational or irrational numbers. Have two statements be true and two statements be false. 24.

25.

26.

27.

10

6.1D Homework: Approximating the Value of Irrational Numbers* Name:

Period:

On the problems below, a calculator may be used for multiplying or squaring only. Do not use the square root button, except to verify your answers. 1. Between which two integers does √

lie? Which integer is it closest to?

a. Show its approximate location on the number line below. Scale your number line.

b. Without using the square root button on a calculator, find the value of √ to the nearest tenth. Show its approximate location on the number line below. Scale your number line.

c. Without using the square root button on a calculator, find the value of √ to the nearest hundredth. Show its approximate location on the number line below. Scale your number line.

2. Use your work from above to approximate the given values to the nearest whole number, nearest tenth, and nearest hundredth. √

a.

b.

√

c. √ 11

Directions: Without using the square root button on a calculator, determine which of the two numbers is greater. Justify your reasoning. 3.

√

4. √

5.

√

or

or √

√

or

6.

√

7.

or 8

8.

9.

10.

or

√

√

or 7

or 6.2

or 10

Directions: Use the given calculations to answer the questions below. 11. Order the following numbers from least to greatest: √

12. Find a number between

nd

13. Find a number between 3.1 and √

̅

Calculations:

̅.

. 12

(problems #14 and #15 are adapted from illustrativemathematics.org)

14. Cindy’s c lcul tor gives why or why not.

v lue of

for . Is the equation

= 3.14159265 valid? Explain

15. When Cindy computes on her calculator, using the and square buttons, it shows 9.86960440. On the other hand, when she calculates on her calculator it shows 9.86960438. Explain why the calculator shows different answers for what appears to be the same quantity.

Directions: Solve the following problems. Do not use the square root key on your calculator. 16. A hospital has asked a medical supply company to manufacture intravenous tubing (IV tubing) that has a minimum opening of 7 square millimeters and a maximum opening of 7.1 square millimeters for the rapid infusion of fluids. The medical design team concludes that the radius of the tube opening should be 1.5 mm. Two supervisors review the design te m’s pl ns, e ch using a different estimation for . Supervisor 1: Uses 3 as an estimation for

Supervisor 2: Uses 3.1 as an estimation for

The supervisors tell the design team that their designs will not work. The design team stands by their plans and tells the supervisors they are wrong. Who is correct and why? (recall that the formula for the area of a circle is )

13

17. A square field with an area of 2,000 square ft. is to be enclosed by a fence. Three contractors are working on the project and have decided to purchase slabs of pre-built fencing. The slabs come in pieces that are 5-ft. long. 

Keith knows that √ is between 40 and 50. Trying to save as much money as possible, he estimates on the low side and concludes that they will need 160 feet of fencing. Therefore, he concludes they should purchase 32 slabs of the material.



Jose also knows that √ is between 40 nd 50 but he is fr id th t using Keith’s c lcul tions, they will not have enough fencing. He suggests that they should estimate on the high side and buy 200 feet of fencing to be safe. Therefore, he concludes they should purchase 40 slabs of material.

Keith nd Jose begin to rgue. S m jumps in nd s ys, “I h ve w y to m ke you both happy – we will purch se enough m teri l to enclose the entire field nd we will minimize the mount of w ste.” Wh t do you think S m’s suggestion is nd how m ny sl bs will be purch sed using S m’s r tion le?

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6.1E Homework: Simplifying Square Roots* Name:

Period:

Directions: Simplify the following square roots. Assume all variables represent positive numbers. 1.

√

2.

√

3.

√

4. √

5.

√

7.

√

8.

9.

√

10. √

11.

√

12. √

6. √

13. √

14. √

√

16. √

15.

17.

√

√

18. √

15

Spiral Review: 19. Write the equation of a line in slope intercept form that is parallel to point (-1, -4)

and passes through the

20. Does the table of values represent a linear situation? Explain why or why not. Day

Height (inches)

5

30

8

76.8

14

235.2

21. Solve for x:

22. Solve the system of linear equations:

23. Amy has $1.15 in her wallet, which contains 18 coins in nickels and dimes. How many of each kind of coin does Amy have in her wallet?

16

6.2A Homework: A Proof of the Pythagorean Theorem Name:

20

18

16

Period:

Directions: In each of the problems below, a right triangle is shown in gray. The squares along each of the three sides of the triangles have been drawn. The area of two of the squares is given. Determine the area of the third square to find the value of the legs and hypotenuse for each triangle. Write your values below each picture. Verify your side lenghs using the Pythagorean theorem. 1.

2.

14

1

12

1

10

4 16

8

6

4

legs:

hypotenuse:

legs:

Verify using the Pythagorean Theorem:

3.

hypotenuse:

Verify using the Pythagorean Theorem:

4.

13

16

9

legs:

hypotenuse:

Verify using the Pythagorean Theorem:

8

legs: hypotenuse: Verify using the Pythagorean Theorem:

17

Directions: For each of the following problems, the gray triangle is a right triangle. Draw the squares adjacent to each of the three sides of the triangles. Find the area of each square and write the area in each square. Then, find the side lengths a, b, c of each triangle. Verify your side lenghs using the Pythagorean theorem. 5.

6.

a = ______

b = ______

c = ______

Verify using the Pythagorean Theorem:

a = ______

b = ______

c = ______

Verify using the Pythagorean Theorem:

Directions: Without a ruler, use your knowledge of right triangles and the Pythagorean Theorem to construct the following objects. Clearly label each object and its vertices. Write the slope for each line segment. 7. Square

that has an area of 5 square units

9. ̅̅̅̅ that has a length of √

units

8. Square units

that has an area of 29 square

10. ̅̅̅̅ that has a length of √

units

18

11. Without a ruler, draw three different squares with an area of 25 square units on the grid below (your squares can not be tilted the same way.)

Spiral Review: 12. Solve:

13. Write the equation of a line in slope intercept form that is perpendicular to through (1,1)

and passes

14. Complete the equation so it has infinitely many solutions:

19

15. Graph and state the y-intercept. Is the line proportional? Explain why or why not.

16. Julie is three times s old s her son. In 12 ye rs, Julie’s ge will be one ye r less th n twice her son’s age. How old is Julie now?

17. Graph

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6.2B Homework: The Pythagorean Theorem and Unknown side Lengths* Name:

Period:

Directions: Two side lengths of a right triangle have been given. Solve for the missing side length, if a and b are leg lengths and c is the length of the hypotenuse. Leave your answer in simplest radical form. 1. a = 16, b = 30, c = ?

2. a = 40, b = ?, c = 50

3. a = 2, b = 2, c = ?

4. a = ?, b = √ , c = 8

Directions: Determine if the given three side lengths form a right triangle. Justify by using the Pythagorean theorem. Show all of your work. 5.

8.6, 14.7, 11.9

6.

7.

8, √ , 8

8.

√ , 8 , 16

7, 11.4, 9

Directions: Find the value of x using the Pythagorean Theorem. Leave your answer in simplest radical form. Show all of your work. 9.

10.

x

7.5

x=

4.5

4

4 x

x=

21

11.

12. 2

x x

x

5 3 5

x=

13.

x=

14.

x

1 2 0.4

0.41

x

x=

x=

16.

15.

x

x

12

x=

8

8

12

x=

22

17. A football field is 360 feet long and 160 feet wide. What is the length of the diagonal of a football field assuming the field is in the shape of a rectangle? Round to the nearest foot.

18. For the right triangle below, what could the lengths of the legs be so that their lengths are integers and x is an irrational number between 5 and 7?

19. Megan was asked to solve for the unknown side length in the triangle below. Her work is shown below. She made a mistake when solving. Explain the mistake she made and then solve the problem correctly. 5

Meg n’s Solution:

Correct Solution:

13

√

Explain Mistake:

20.

Raphael was asked to solve for the length of the hypotenuse in a right traingle with legs that have side lengths of 4 and 5. His work is shown below. He made a mistake when solving. Explain the mistake and then solve the problem correctly. R ph el’s Solution:

Correct Solution:

Explain Mistake: 23

21.

You are locked out of your house. You can see that there is a window on the second floor that is open so you plan to go and ask your neighbor for a ladder long enough to reach the window. The window is 20 feet off the ground. There is a vegetable garden on the ground below the window that extends 10 ft. from the side of the house th t you c n’t put the l dder in. Wh t size l dder should you sk your neighbor for?

22.

Ray is a contractor who needs to ccess his client’s roof in order to ssess whether the roof needs to be replaced. He sees that he can access a portion of the roof that is 15 feet from the ground. He has a ladder that is 20 feet long. a. How far from the base of the house should Ray place the ladder so that it just hits the top of the roof? Round your answer to the nearest tenth of a foot.

b. How far should he place the ladder from the base of the house if he wants it to sit 3 feet higher than the top of the roof? Round your answer to the nearest tenth of a foot.

24

6.2C Homework: Distance Between Two Points* Name:

Period:

Directions: Use the distance formula to find the distance between the two points shown on each grid. Leave your answers in simplest radical form. Justify your answer by drawing a right triangle on the grid, finding the distance of the other legs and applying the Pythagorean Theorem. 1.

2.

Distance of ̅̅̅̅ =

Distance of ̅̅̅̅ =

Justification with Pythagorean Theorem:

Justification with Pythagorean Theorem:

Directions: Find the distance between the two points given below. Leave your answers in simplest radical form. Show all of your work. 3.

4.

25

5.

6.

7.

8.

9.

10.

(

)

(

)

26

11. Determine if triangle ABC with vertices A(-3,4), B(5,2) and C(-1,-5) is an isosceles triangle. .

12. Three right triangles surround a shaded triangle; together they form a rectangle measuring 12 units by 14 units. The figure below shows some of the dimensions but is not drawn to scale. Is the shaded triangle a right triangle? Provide a proof for your answer.

27

6.2D Homework: Creating Cube Roots* Name:

1. Does √

Period:

√

. Explain why or why not.

2. Complete the table below. Write your answers as simplified radicals. Side length Volume

0.2 40

s

96

V

Directions: Simplify. Show all of your work. Assume variables represent positive numbers. 3. √

4.

√

5.

√

√

6.

√

7.

√

8.

9.

√

10.

√

11.

√

28

12.

√

15. √

13. √

14.

√

16. √

17.

√

Spiral Review: 18. Solve:

19. Solve the system of linear equations:

20. In seven years, Patti will be twice as old as Deena will be. Deena is one-third P tti’s ge now. What are their current ages?

29

6.2E Homework: Solve Equations using Square and Cube Roots* Name:

Period:

1. Explain why taking the cube root of a rational number does not produce two answers. Provide an example to justify your reasoning.

Directions: Solve each equation. Show all of your work. √

2.

3.

4.

5.

6. √

7.

8.

9. √

10.

30

11.

14.

12.

where p is a positive rational number

13.

15. Solve for r where A is the area of a circle and r is the radius:

17. Write and solve an equation of your own that has a power of 2 in it.

16. Solve for r where V is the volume of a cylinder, r is the radius, and h is the height:

18. Write and solve an equation of your own that has a power of 3 in it.

19. How f r c n you see when you’re t the top of re lly t ll building? How f r c n you see if you’re on top of the Empire State Building? The Sears Tower in Chicago? The furthest distance you can see across flat land is a function of your height above the ground. The function is √ , where h is height in meters of your viewing place and d is distance in kilometers you can see. a. The equation above can be used to find the distance when you know the height. Rewrite the equation to find height when you know the distance.

31

b. If you were lying down on top of a building that is 100 meters tall, how far could you see? Write an equation that represents this situation and solve. Show all your work.

c. The CN Tower in Toronto, Canada is 555 meters tall. It is near the shore of Lake Ontario, about 50 kilometers across the lake from Niagara Falls. Your friend states that on a clear day, one can see as far as the falls from the top of the Tower. Are they correct? Explain your answer.

d. The Washington Monument in Washington D.C. is 170 meters tall. How far can one see from its top? Write an equation and solve. Show all your work.

e. How high must a tower be in order to see at least 60 kilometers? Write an equation and solve. Show all your work.

f. Advertising for Yahoo Amusement Park claims you can see 40 kilometers from the top of its observation tower. How high is the tower? Write an equation and solve. Show all your work.

32

20. To enhance understanding of the relation between height and viewing distance, complete the table below. Express each output value to the nearest tenth, then plot the points on the graph. Do not connect the points. Height (m)

0

50

100

150

200

250

300

350

400

450

500

Distance (km)

21. Does the relationship between height and viewing distance represent a linear function? Explain why or why not.

33