Rational and Irrational Numbers Math B Honors

Module #6 Homework 2017-2018

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. List the first 12 perfect square numbers. 2. What is the side length of a square with an area of 0.0036 units2? 3. What is the area of a square with a side length of

units?

4. What is the side length of a square with area 35?

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

6. Show and explain two different methods for finding the area of the shape below.

Method #1

Method #2

7. Create a square with area 32. Every vertex must be on a dot. Explain your process.

SDUHSD Math B Honors Module #6 – HOMEWORK 2017-2018

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

9. √

10.

√

11. √

12. √

13. √

14. √

15. √

16. √

17. 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.

Spiral Review: 18. Solve:

19. Write a linear equation in standard form that represents the table of values.

20. Without using a calculator, convert

x

0

100

200

300

y

1.5

36.5

71.5

106.5

to a decimal. Show all of your work.

SDUHSD Math B Honors Module #6 – HOMEWORK 2017-2018

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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.

7.

8.

9.

10.

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

12.

SDUHSD Math B Honors Module #6 – HOMEWORK 2017-2018

3

13.

̅̅̅̅

14.

̅

15.

̅

16.

̅̅̅̅

Directions: Simplify. Express answers as simplified fractions. 17.

̅

18.

̅

19.

̅

Spiral Review: 20. A line passes through the points (6, 3), (-4, -2) and (n, 1). What is the value of n?

21. 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?

SDUHSD Math B Honors Module #6 – HOMEWORK 2017-2018

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6.1C Homework: Expanding Our Number System* Name:

Period:

Directions: Classify each of the following numbers by writing an “X” under the set(s) to which it belongs. Number

Natural number

Whole number

Integer

Rational number

Irrational number

Real

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 361

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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

15. integers from -1 to 2

16. real numbers greater than -2

17. whole numbers less than 0

Directions: Determine if the given statement is true or false. If false, re-write the statement to make it true. 18. The number

is rational.

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

20. The number

̅ is irrational because its decimal expansion goes on forever.

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

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Directions: Use your knowledge of the real number system to answer the following questions. 22. When can a number be a whole number, an integer, and a rational number? Explain your reasoning.

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

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

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

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

27. Create three statements about rational or irrational numbers. Have one statement be true and two statements be false. True Statement

False Statement

SDUHSD Math B Honors Module #6 – HOMEWORK 2017-2018

False Statement

7

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 closer to?

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

2. Use your work from above to approximate the value of each expression using whole numbers, tenths, and hundredths. a.

√

b.

√

c. √

whole numbers:

whole numbers:

whole numbers:

tenths:

tenths:

tenths:

hundredths:

hundredths:

hundredths:

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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

and

13. Find a number between 3.1 and √

Calculations:

̅

̅.

.

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

14. Cindy’s calculator gives a value of why or why not.

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.

Spiral Review: 16. Complete the equation so that it has no solution: (

)

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

18. 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?

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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.

√

6. √

7.

√

8.

9.

√

10.

√

SDUHSD Math B Honors Module #6 – HOMEWORK 2017-2018

√

11

11.

√

12.

13.

√

14. √

15.

√

16. √

17.

√

√

18. √

19. √ ̅

20.

21. √

22. √

SDUHSD Math B Honors Module #6 – HOMEWORK 2017-2018

√

12

6.1F Homework: Products and Quotients of Square Roots Name:

Period:

Directions: Simplify. Show all of your work. Assume all variables represent positive numbers. Rationalize the denominator when necessary. 1. √

4.

7.

√

√

√

10. √

2. √

√

5. √

√

8.

11.

√

√ √

3.

√

6.

√ √

9. (√

√

SDUHSD Math B Honors Module #6 – HOMEWORK 2017-2018

12.

)

√

√

13

13. Find the circumference of a circle with radius √ simplest radical form.

cm. Write your answer in terms of

14. Find the area of a semi-circle with diameter √ simplest radical form.

meters. Write your answer in terms of

and in

and in

15. Determine the side length of a square with area 0.24 inches. Write your answer in simplest radical form.

16. Find the area of a triangle with a base of 1.5 cm and a height of √ simplest radical form.

cm. Write your answer in

Spiral Review: 17. Write the equation of a line in slope-intercept form that is parallel to the point (-1, -4)

18. Graph

and passes through

and label the boundary line.

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6.1G Homework: Sums and Differences of Square Roots Name:

Period:

Directions: Simplify. Assume all variables represent positive numbers. Rationalize the denominator when necessary. √

1.

√

3. √

√

5. √

√

7.

√

√

√

2.

4.

√

6.

8.

√

√

√ √

√ (

√

SDUHSD Math B Honors Module #6 – HOMEWORK 2017-2018

√

)

√ √

15

9.

11.

√

√

√

√

10.

√

√

√

12. √

√

Directions: Tell whether each statement is true or false. If false, explain the mistake that was made and re-write the statement correctly. Assume all variables represent positive numbers. 13.

√

15.

√

√

√

14. (√

16.

17. Determine the area of a square whose perimeter is

)

√

√

√ meters.

18. Determine the perimeter of a square whose area is 252 meters squared.

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Section 6.1: Review Name:

Period:

Directions: Simplify the following. Assume all variables are positive. Rationalize the denominator when necessary. √

1. √

2.

4. √ ̿

5. √

√

7. √

10.

√ √

8.

3.

(√

√

11. √

√ )

√

6. √

√

9.

√

√

√

√

SDUHSD Math B Honors Module #6 – HOMEWORK 2017-2018

12.

√

√

√

√ √

17

13. √

14.

√

15. √

16. Create a square with area 8. Every vertex must be on a dot.

Directions: Classify the given number. Number

Natural number

Whole number

Integer

Rational number

Irrational number

Real

17. 18.

0

19. 20.

√ ̅

21. 22.

√

23.

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Directions: Determine if the given statement is true or false. If false, re-write the statement to make it true. 24. All irrational numbers are real.

25. The square root of any number is irrational.

26. All integers are whole numbers. .

Directions: Without a calculator, change each decimal to a fraction. Show all of your work. 27.

29.

̅

̅

28.

̅

30.

̅̅̅̅

Directions: Represent the given set on a number line. 31. integers from -1 to 3

32. real numbers from -4 to 5

33. natural numbers from -2 to 3

34. Give an example of a rational number between √ and √

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35. Give an example of an irrational number between 8 and 9.

36. Without using a calculator, graph the given values on the number line. √

√

√

Direction: Use the calculations to answer the questions below. ̅ from least to greatest.

37. Order the numbers √

38. Find a number between 1.4 and

Calculations:

.

39. Would 2.24 be located to the right or to the left of √ on the number line?

40. Without using the square root button on a calculator, approximate √

to the specification given.

a. Nearest whole number

b. Nearest tenth

c. Nearest hundredth

41. Order the following numbers from least to greatest. √

SDUHSD Math B Honors Module #6 – HOMEWORK 2017-2018

√

20

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

20

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 lengths using the Pythagorean Theorem.

18

16

1.

2.

14

1

12

1

10

4 16

8

6

legs:

4

legs:

hypotenuse:

hypotenuse: Verify using the Pythagorean Theorem:

Verify using the Pythagorean Theorem:

3.

4.

13

16

9

legs:

8

hypotenuse: legs:

hypotenuse:

Verify using the Pythagorean Theorem: Verify using the Pythagorean Theorem:

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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 lengths using the Pythagorean Theorem. 5.

6.

a = ______

b = ______

c = ______

Verify using the Pythagorean Theorem:

a = ______

b = ______

c = ______

Verify using the Pythagorean Theorem:

7. In using the figure to prove the Pythagorean Theorem, which statement would not be used? A. The area of the smaller square, . c2, is equal to

B. The area of the large square is equal to

C. The four triangles are all congruent.

D. The area of the large square, less the area of the smaller square, is equal to ( )

8. Without a ruler, draw two different squares with an area of 25 square units on the grid below. Your squares can not be tilted the same way.

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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. 9. Square 10. Square

that has an area of 5 square units that has an area of 29 square units

11. ̅̅̅̅ that has a length of √

units

12. ̅̅̅̅ that has a length of √

units

Spiral Review: 13. Complete the equation so that it has infinitely many solutions:

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

and passes

15. Julie is three times as old as her son. In 12 years, Julie’s age will be one year less than twice her son’s age. How old is Julie now?

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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. If a and b are leg lengths and c is the length of the hypotenuse, solve for the missing side length. Leave your answer in simplest radical form. 1. a = √

2. a = 2, b = ?, c = √

, b = 4, c = ?

4. a = ?, b = √ , c = 8

3. a = , b = , c = ?

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

7.

8.6, 14.7, 11.9

,

6.

√ ,

1.2, 1.6, 2

8.

,

,

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

c

10. 2

6

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11.

12.

x

0.4

0.41

13. Find the value of x in the diagram. Write your answer in simplest radical form

14. Raphael was asked to solve for the length of the hypotenuse in a right triangle 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. Raphael’s Solution:

Correct Solution:

Explain Mistake:

What advice can you give Raphael to avoid this mistake in the future?

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15. You are locked out of your house. You can see that there is a window on the second floor that is open and you plan to 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 and you cannot put the ladder in the vegetable garden. What size ladder should you ask your neighbor for?

16. A new restaurant is putting in a wheelchair ramp. The landing that people enter the restaurant from is 2 feet higher than street level. Safety standards require that for every 1 foot of rise on a wheelchair ramp there must be a minimum run of 12 feet. a. How long will the ramp have to be to meet safety standards? Round your answer to the nearest tenth.

b. Will a 30 foot long ramp work? If yes, how far would the ramp be from the base of the landing? Round your answer to the nearest tenth.

Spiral Review: 17. Seth has a total of 42 dimes and quarters. If the total value of all the coins is $8.25, how many dimes and quarters does Seth have?

18. Does the table of values represent a linear relationship? If so, write the equation in slope-intercept form.

SDUHSD Math B Honors Module #6 – HOMEWORK 2017-2018

x

y

1

5

3

9

10

23 26

6.2C Homework: Special Right Triangles Name:

Period:

Directions: Use the relationships in special right triangles and the Pythagorean theorem to find the length of the missing sides for each triangle. Write lengths in simplest radical form. 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

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11. The perimeter of a square is 42 inches. Find the length of the diagonal.

12. A square has a diagonal with a length of √ inches. What is the side length of the square?

13. An equilateral triangle has a side length of 15 inches. Find the length of the triangle’s altitude

14. Find all the missing side lengths in the picture.

15. A baseball diamond is in the shape of a square. The distance between each of the consecutive bases is 90 feet. What is the distance from Home Plate to 2nd Base? Round your answer to the nearest whole number.

16. The cube below has a diagonal ( ) length of 15 cm. What is the length of each side of the cube? Show all of your work. Write your answer in simplest radical form.

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6.2D 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.

5.

6.

(

SDUHSD Math B Honors Module #6 – HOMEWORK 2016-2017

)

(

)

28

7. The point (4, 8) lies on a circle centered at (12, 14). a. Use the distance formula to find the diameter of this circle. Show all of your work.

b. Find the circumference. Round your answer to the nearest tenth. Show all of your work.

c. Find the area of the circle. Round your answer to the nearest tenth. Show all of your work.

8. Determine if triangle ABC with vertices A(-3,4), B(5,2) and C(-1,-5) is an isosceles triangle. Use the coordinate plane as needed.

9. Find the perimeter of square QRST if two of the vertices are Q(6,7) and R(-3,4). Write the perimeter in simplest radical form. Use the coordinate plane as needed.

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10. Use the distance formula and the Pythagorean Theorem to show that triangle ABC with vertices at A(3,-2) and B(-3,7) and C(-9,3) is a right triangle.

11. 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.

Spiral Review: 12. Solve:

13. Solve:

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6.2E Homework: Creating Cubes and nth Roots* Name:

Period:

1. Explain the error that was made below when simplifying √ correctly. √

√

. Then simplify the expression

√

√

√ √

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

0.2 40

s

96

V

Directions: Simplify. Assume variables represent positive numbers. 3. √

4.

√

5.

√

√

6.

√

7.

√

8.

9.

√

10.

√

11.

SDUHSD Math B Honors Module #6 – HOMEWORK 2016-2017

√

31

12. √

13. √

14.

√

15. √

16. √

17.

√

18. √

19.

√

20.

√

Spiral Review: 21. Solve:

22. Solve the system of linear equations:

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

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6.2F Homework: Solve Equations using Square and nth 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. Express your answer in simplest radical form. Show all of your work. 2.

5.

3.

√

6.

√

√

4.

7.

Directions: Directions: Solve the equation. Express your answer in simplest radical form and check your solution. Show all of your work. 9. √

8.

Check:

Check:

SDUHSD Math B Honors Module #6 – HOMEWORK 2016-2017

10.

√

Check:

33

11.

12.

Check:

Check:

14. √

Check:

Check:

√

√

Check:

16. √

15.

Check:

17. √

13.

Check:

19. √

18.

Check:

SDUHSD Math B Honors Module #6 – HOMEWORK 2016-2017

√

√

Check:

34

Section 6.2: Review Name:

Period:

Directions: Use Pythagorean Theorem to find the length of the missing sides. Write your answer in simplest radical form. Show all of your work. 1.

2.

c 5 11

3.

4.

5. Determine whether the given lengths form a right triangle. Verify your answer. a. 13.5, 5.5, 12.5

6.

b.

, , √

Abby 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 her mistake and then solve the problem correctly. Abby’s Solution:

Correct Solution:

8 x

x

Explain Mistake:

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7. One side length of a rectangle is 2 inches. The diagonal of the rectangle has a length of √ inches. What is the length of the other side of the rectangle? .

√ meters.

8.

Determine the area of a square whose perimeter is

9.

The dimensions of a kite sail are shown below. The support rod that runs from the top of the kite to the bottom of the kite has been broken and needs to be replaced. What length of rod is needed to replace the broken piece? Round your answer to the nearest tenth. 52 cm 42 cm 42 cm

77 cm

Directions: Find the distance between the given points. Write your distance in simplest radical form. Show all of your work. 10. (-3,8) and (5,4)

11. (5,2) and (3,10)

12. (

) and (

)

13. Given that (1, -2), (0, 5) and (-3,1) are the vertices of a right triangle, determine an equation of the line that passes through the endpoints of the hypotenuse. Write your equation in standard form.

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Directions: Determine if the given statement is true or false. If false, re-write the statement to make it true. 14. The sum of two rational numbers is rational.

15. Between two rational numbers there is an irrational number.

16. The sum of two irrational numbers, that are not opposites, is irrational.

Directions: Solve the equation. Express your answer in simplest radical form. Show all of your work. √

17.

18.

19. √

20. √

√

Directions: Solve the equation. Express your answer in simplest radical form and check your solution. Show all of your work. 21.

22.

Check:

Check:

24. √

23.

Check:

√

Check:

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