ms. campos unit 12 real number system and radicals 2017-2018 · operations with rational and...
TRANSCRIPT
1
Ms. Campos – Unit 12
Real Number System and Radicals
2017-2018
Name _______________________________
Date Lesson Topic Homework
X M 4/9 1 Perfect Squares and Perfect Cubes Lesson 1 -Page 6
X T 4/10 2 Rational vs Irrational No Homework
2 W 4/11 3
Operations with Rational and Irrational
Numbers (ELA)
No Homework
3 T 4/12 4
Estimating, Comparing and Ordering
Square Roots (ELA)
Lesson 4 – Pages 17
4 F 4/13 5 Simplest Radical Form Lesson 5 – Page 20
5 M 4/16 6 Simplest Radical Form Lesson 6 – Page 23
6 T 4/17 Review Finish Review Sheet
1 W 4/18 Test
2
Unit 12 – Lesson 1
Aim: I can identify the first 15 perfect squares
Warm Up: Round the following to the nearest whole number:
1) 159.513 2) 2,374.928 3) 4.1
4) 0.783 5) 19.51 6) √55
Round the following to the nearest tenths:
7) 159.513 8) 2,374.928 9) 4.18
10) 0.783 11) 12.63 12) √87
Guided Practice:
List the first 15 perfect squares:
1) 6) 11)
2) 7) 12)
3) 8) 13)
4) 9) 14)
5) 10) 15)
Simplify
1) √36 = ________ because ( )2 is 36 2) −√64 = ________
3) ±√. 09 = ________ 4) √81
9 = ________ 5) −
√196
√25 = ________
6) √121 7) √25 8) -√49 9) ±√100
3
Non-Perfect Square Roots
Non-Perfect square
Round to the nearest tenth
10) √8 11) √108 12) −√6 13) √150 14) √80
9
Perfect Cubes
List the first 10 perfect cubes:
Simplify
15) √273
= because ( )3 is 27 16) − √8
3
17) ± √125 3
18) √216
27
3 19) −
√3433
√49
Non-Perfect Cubed Roots
Non-Perfect cubes
Round to the nearest tenth.
20) √123
21) −√603
22) √1003
23) √803
24) −√753
4
Equations: Solve each equation for x
1) 𝑥2 = 81 2) 𝑥3 = 1000 3) 𝑥2 = 1.44 4) 𝑥2 = 121 5) 𝑥3 = 512
6) 𝑥3 = 125 7) 𝑥2 = 36 8) 𝑥2 = 1 9) 𝑥2 = 144
11) 𝑥3 = .064 12) 𝑥3 = 1000 13) x3 = 1
8
10) The area of a square boxing ring is 225ft2. What is the length of one side? (a = s2)
17) The volume of a cube is 125 ft3. What is the length of one side? ( v = s3)
18) Alex just built a pool in his backyard. He needs to put up a fence around the pool. The area he
needs to enclose is a square with an area of 229 square feet.
a) What is the length of each side of the square that Alex wants to enclose to the nearest
hundredth?
b) What is the total amount of fence that Alex needs to put up?
5
Problem Set:
1) What is the value of √100?
A. 4
B. 10
C. 25
D. 50
2) What is the value of √273
?
A. 3
B. 5
C. 9
D. 13.5
3) Solve for x: x2 = 256
A. x = ±16
B. x = ±15
C. x = ±16
D. x = ±128
4) Solve for y: y3 = 216
A. y = 4
B. y = ±4
C. y = 6
D. y = ±6
5) Which statement below is true?
A. √1 = √13
B. √2 = √33
C. √4 = √93
D. √4 = √273
6) Which statement below is true?
A. √4 = √43
B. √4 = √273
C. √16 = √273
D. √16 = √643
7) Robert thinks the cube root of 1,000 is 10.
Is he correct? Explain how you know.
____________________________________
___________________________________
____________________________________
8) The floor of a square bedroom has an area of
169 square feet.
Part A: What is the length of each side of the
bedroom? _______
Part B: The floor of a square family room has an area
twice as great as the area of his bedroom
floor. Is the length of each sid of the family
room twice as great as the length of each side
of the bedroom floor?
Explain how you know.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
9) The volume of a cube is 512 cm3.
Part A: What is the length of one side? _________
Part B: What is the area of one of the faces? _______
6
Unit 12 -Lesson 1 Homework
Evaluate each square root or cubed root.
1) √225 2) - √49 3)√81 4) ±√121
5) √144 6) √27
3 7) −√81 8)
√225
25 9) √
64
125
3
Solve each equation.
10) p3 = 343
1000 11) 𝑥2 = 4 12) 𝑥3 = 64 13) 𝑥2 = 36 14) 𝑥3 = 125
15) The volume of a cube is 1,000 ft3. What is the length of each side of the cube?
16) Alex just built a pool in his backyard. He needs to put up a fence around the pool. The area he
needs to enclose is a square with an area of 225 square feet.
a) What is the length of each side of the square area that Alex wants to enclose?
b) What is the total amount of fence that Alex needs to put up? (Hint: Perimeter)
Round to the nearest tenth
17) √604
12
3 18) √65
3 19) −√6 20)
√160
√4 21) √
36
6
3
22) The volume of a cube is 1,500 ft3. What is the length of each side of the cube to the nearest
hundredth?
7
Unit 12- Lesson 2
Aim: I can identify the different number sets that make up the Real Number System
Warm Up:
1) What is the value of √16?
A. 4
B. 10
C. 25
D. 50
2) What is the value of √373
to the nearest
whole number?
A. 4
B. 3
C. 2
D. 3.3
3) Find the radicand: √3
= 6
A. 125
B. 216
C. 343
D. 512
4) Approximate the value of x to the nearest
tenth: x2 = 250
A. x = ±15.81
B. x = ±15
C. x = ±15.9
D. x = ±15.8
5) Which of the following has the correct
approximation to the nearest tenth?
A. √10 = 3.1
B. √20 = 4.4
C. √40 = 6.3
D. √45 = 6.8
6) Which of the following has the correct
approximation to the nearest whole number?
A. √2003
= 6
B. √73
= 1
C. √863
= 5
D. √523
= 3
7) Mateo thinks the cube root of 120 is approximately
5. Is he correct? Explain how you know.
____________________________________
____________________________________
____________________________________
8) The floor of Bobby’s square bedroom has an area
of 147 square feet.
Part A: What is the length of each side of Bobby’s
to the nearest tenth? _______
Part B: The floor of Tommy’s square bedroom
has an area of 180. How many feet longer is
Tommy’s bedroom floor. _______
9) The volume of a cube is 550 cm3.
Part A: What is the length of one side to the nearest
whole number? _________
Part B: What is the area of one of the faces?
8
Guided Practice:
Real Number System
Counting Numbers (Natural Numbers):
Whole Numbers:
Integers:
Rational:
Irrational:
Real Numbers are made up of __________________________and ___________________________ numbers.
Rational Numbers Irrational Numbers
1) 1)
2)
2)
3)
3)
4)
9
Tell whether each is Rational or Irrational and why.
1) .424242... 2) -.5638 3) . 68
4) -3.312334... 5) -.000009 6) 4
5
7) 34
53454 8) 0 9) 456,812
10) 3.14 11) 7𝜋 12) 7
22
13) √4
81 14) √85 15) √625
Extended Response
16) a) Circle the rational number? 𝜋 √8 √81
b) Explain why the number(s) you circled is/are rational.
___________________________________________________________________________________
___________________________________________________________________________________
17) a) Circle the irrational number? √85 5
√16
√4
√8
b) Explain why the number you chose is irrational.
__________________________________________________________________________________
___________________________________________________________________________________
18) Circle only the rational numbers
7.8 3.8732... 5. 7 -17
√361 √300 12𝜋 32
16
10
Problem Set:
19) Which letter on the number line below best represents the value of √6?
0 1 2 3 4 5 6
20) Place the following numbers on the number line:
√5 ,3
2 , −0.5 , −√4 , 𝜋
-2 -1 0 1 2 3 4
Tell whether each number is rational or irrational.
21) √21 _________________________ 22) √5 __________________________
23) -7 ___________________________ 24) 22 __________________________
25) 0_____________________________ 26) √46 __________________________
27) 4.5 ___________________________ 28) 7. 3̅ __________________________
29) 3
7 ____________________________ 30) 4
23 ___________________________
31) 8 ____________________________ 32) 0.7____________________________
33) 1. 6̅ __________________________ 34) -0.333 _________________________
35) 0.30729...______________________ 36) 7
22 __________________________
37) √50__________________________ 38) 5.101001... _____________________
39) 9 𝜋 _________________________ 40) 2.7913... _______________________
41) 4.8936..._______________________ 42) 2. 28̅̅̅̅ __________________________
43) √12__________________________ 44) 3. 3̅ ___________________________
B A D C
11
Unit 12 – Lesson 2 Homework
Choose the best answer:
1) Which of the following is an integer, but not a
whole number?
A) 0 B) -11 C) 0 D) -5.5
2) Which of the following is not a counting number
(natural number)?
A) 0 B) 1 C) 2 D) 3
3) Which is an example of a whole number?
A) 0 B) -11 C) 2.5 D) 1
2
4) Which number is an integer, a whole number, and
a counting (natural) number?
A) 0 B) -1 C) 15 D) 0.5
5) Which of the following is a rational number?
A) 𝜋 B) −11
C) √2 D) −5.5234 …
6) Which of the following is irrational?
A) 0 B) 1
2
C) 2𝜋 D) 3.14
7) Which of the following is an example of a
non-perfect square?
A) √9 B) √81 C) √225 D) √45
8) What is the first counting (natural) number?
A) 0 B) 1 C) -1 D) 0.1
9) Which of the following is not a rational number?
A) 6 B) 1
3 C) 7𝜋 D) 3.5
10) Which of the following is a rational number?
A) 7.2348 … B) 𝜋 C) √25 D) 8
9
11) Which of the following is not an irrational
number?
A) 4 B) 𝜋
3 C) √7 D) √200
12) Which of the following is an irrational number?
A) 9.5 B) 3.14 C) 9𝜋 D) −0.5
13) √197 lies between which two consecutive
integers?
A) 196 & 197 B) 14 & 15
C) 15 & 16 D) 197 & 198
14) Jessica is asked if 3𝜋 is a rational number.
Which of the following is the most logical
response?
A) “No, it is irrational because any multiple
of 𝜋 is irrational.”
a) B) “Yes, it is rational because 3𝜋 can be written as
b) a fraction.”
c) C) “No, it is irrational because 3 is a prime
d) number.”
e) D) “Yes, it is rational because 3 and 𝜋 are both
f) rational.”
g)
h) 15) Real numbers are made up of _______________
i) and __________________ numbers.
12
Unit 12 – Lesson 3
Aim: I can determine if the sum/difference or product/quotient of 2 Real numbers is rational or irrational
Warm Up:
5. Given the list of numbers below, determine which number sets they belong to.
√18 ,2
9 , −0.75 , −√4 , 5𝜋
a) Integers ______________ b) Rational _________________ c) Irrational _____________
13
Guided Practice:
Operations with Real Numbers
First
Number
Second
Number Sum
Is the Sum
Rational or
Irrational?
Product
Is the
Product
Rational or
Irrational?
1
9
2
7
-7 2𝜋
√5 √5
√4 √5
−√5 √5
𝜋 −𝜋
0 √12
Sum/Difference Product/Quotient
14
Problem Set:
1. True or False: If an expression contains π, the expression is irrational.
2. True or False: Since the number √2
7 is a fraction, it is a rational number.
3. Is √18
2 rational or irrational? Explain.
4. Is the sum of 5.8 and 7π rational or irrational? Justify your reasoning.
5. Is 4 + 3√24 rational or irrational? Explain.
6. For which value of A and B is A + B a rational number?
1. A = 1
√5 and B =
1
√10 3. A =
1
√3 and B =
1
√8
2. A = 1
√16 and B =
1
√25 4. A =
1
√36 and B =
1
√2
7. Which expression results in a rational number?
1. 1.5 + √1.5
2. 12 − √12
3. 𝟑
𝟒× √
𝟑
𝟒
4. 25 ÷ √25
15
Unit 12 - Lesson 4
Aim: I can estimate, compare, and order Square Roots
Warm Up: Write the inequality symbol for each:
Less Than Greater Than Less than or equal to Greater than or equal to
Guided Practice:
Understand Square Roots and Cubed Roots
Find the two consecutive whole numbers the square root lies between.
1) √3 2) √7 3) √111 4) −√50 5) −√27
Compare: Use >, <, ≤, ≥, or =
6) 12 √12 7) √169 13 8) -10 - √81 9) √273
√12
10) Part A: Put in order least to greatest Part B: Place the original numbers on the number line:
√9 , √5 , 2 , 1.2 , √24 , √1253
0 1 2 3 4 5
Find the two consecutive integers the square root lies between.
11) √29 12) √110 13) −√15 14) −√72
Compare: Use >, <, ≤, ≥, or =
15) 16 √16 16) √121 11 17) -16 -√225 18) √83
√45
16
Problem Set:
Order from least to greatest
19) √2 , 5.85, 42, − √33 20) √125 , √83
, −√100 , -√201
21) Which of the following is the best estimate for √58 ?
A) 2.9 B) 7.7 C) 7.6 D) 5.8
22) Place the following numbers on the number line:
√2 , √18 , 0.5 , √4 , √11
0 1 2 3 4 5
23) Between which two whole numbers does √89?
A. between 4 and 5
B. between 7 and 8
C. between 8 and 9
D. between 9 and 10
25) Which radical when rounded to the nearest tenth is closest in value to 8?
A. √45
B. √55
C. √65
D. √75
24) Between what two consecutive whole
numbers is √75?
A. 5 and 6
B. 6 and 7
C. 7 and 8
D. 8 and 9
17
Unit 12 - Lesson 4 Homework
Find the two consecutive integers the square root lies between.
1) √19 2) √99 3) √76 4) √215 5) √181
6) √226 7) √42 8) −√6 9) −√55 10) −√77
Compare: Use >, <, ≤, ≥, or =
11) 7 √7 12) √121 11 13) -16 -√225 14) √83
√45 15) 3 √9
Order from least to greatest
16) √125 , √83
, √100 , -√201 17) √224, 103 √57 , 5.89
18) Which of the following is the best estimate for √26 ?
A) 25 B) 4.9 C) 6.2 D) 5.1
19) Which of the following is the best estimate for √79 ?
A) 8.9 B) 9.5 C) 8.1 D) 7.9
20) Which of the following is the best estimate for √30 ?
A) 5 B) 6 C) 5.5 D) 6.5
21) Which of the following is the best estimate for √50 ?
A) 7.1 B) 6.9 C) 5 D) 10
22) Place the following numbers on the number line:
√9 , √5 , 2 , 1.2 , √24 , √1253
0 1 2 3 4 5
18
Unit 12 – Lesson 5
Aim: I can simplify radicals (Day 1)
Warm Up:
1. What are the first 12 perfect squares?
2. Are perfect squares rational or irrational?
3. Give an example of an irrational square root.
4. The expression 98 is a number between
(1) 49 and 50 (2) 9 and 10 (3) 97 and 99 (4) 81 and 100
5. Simplify ( )27
Guided Practice:
Simplifying Square Root Radicals
The symbol √ is called the radical sign and the quantity under the radical sign is the radicand.
For example, in √64 which we read as “the square root of 64” or radical 64.
Express in simplest radical form
When you are asked to simplify a radical:
1.
2.
3.
19
1) 12 2) 64
3) 20 4) √81 5) 8 6) 28
7) √25 8) 27 9) √90 10) √300
11) √16 12) √40 13) √75 14) √45
15) √100 16) 635 17) 108 18)√72
20
Unit 12 - Lesson 5 Homework
Express each in simplest radical form.
1) 24 2) 40 3) √8
4) √99 5) √28 6) 64
7) 12 8) 32 9) 18
10) 45 11) √50 12) 27
13)
14)
21
Unit 12 – Lesson 6
Aim: I can simplify radicals (Day 2)
Warm Up: Compare using <, >, or =:
1) 8 √8 12) √196 14 13) -7 −√64 14) √643
√18
Guided Practice:
Simplifying Radicals
1) 275 2) 243 3) 2 36 4) 184
5) 548 6) 502 7) 4510 8) 492
9) 983 10) 405 11) 94 12) 642
13) 300 14) 452 15) 32 16) 205
22
Problem Set:
17) 5016 18) 812 19) 3√98 20) 6√16
21) Find the two consecutive whole numbers the square root lies between.
a. √2 b. √8 c. √112 d. −√51 e. −√28
22)
23)
24) Put in order from least to greatest, and then place on the number line. √16 , √11, 1, 2.8, −√5 , √273
-2 -1 0 1 2 3 4
Compare using <, >, or =.
25) 5
4
6
5 26) 0.35
3
1 27) 5.02
5
15 28) -8 -√9
Tell whether each description is rational or irrational:
29) Terminating decimals are _____________________________.
30) Pi is _________________________________.
31) Decimals that repeat without a pattern are ____________________________________.
32) Fractions are ________________________________.
33) The square roots of perfect squares are ______________________________________.
34) The square roots of all other positive integers are _______________________________.
35) Decimals that repeat with a pattern are _______________________________________.
23
Unit 12 – Lesson 6 Homework
Examples: Tell whether each is Rational or Irrational.
36) .575757... 37) -.57 38) . 95
39) -3.312334... 40) -.00067 41) 4
7
42) Round √34 to the nearest tenth 43) Approximate −√97 to the nearest tenth.
44) Round ±√203
to the nearest hundredth. 45) Approximate √6383
to the nearest tenth.
46) Round the answer 𝑥2 = 50 47) Round the answer 𝑥3 = 70 to the nearest whole
to the nearest tenth. number.
48) List the first 10 perfect cubes:
Solve: Round to the nearest tenth if necessary
49) 𝑥2 = 81 50) x3 = 8
51) x2 = 75 52) x3 = 450
24
Unit 13 Review Express in simplest radical form:
1. 50 2. 5 72 3. 3√200 4. √20
5. 8 √25 6. √99 7. −3√36 8. 6√27
9. 7√4 10. √8 11. 2
7√49 12. 165
13. Between which two consecutive integers is 75 ?
14. Between which two consecutive integers is 180 ?
15. Which of the following is an irrational number?
A) 4 B) 3. 6̅ C) 2 D) 3.5678
16. What is the length of the side of a square that has an area of 144 cm2? What is the perimeter of the square?
17. Place the following in order from least to greatest: 6, 10 , 2, 8 , 12
18. Solve: Round to the nearest tenth 19. Compare √15 _______√15
3
x3 = 121
25
Identify each as Rational or Irrational
19. 0.24 20. .636363… 21. .128 22. .15847… 23. 1.32332333…
24. 1.32332333 25. 81 26. 14 27. 5 28. 7
1
29. Given E = √3, F = 5√5, G = √4, and H = √1.
Which expression results in a rational number?
1. E + F 2. F + G 3. G + H 4. H + E
30. What is the complement of 25? 31. What is the supplement of 25?
32. Solve the system: 2x + 3y= 20 33. What is the slope of the line 5x + 2y = 12?
5x – 3y = 8
34. Find the slope given the two points (6,-2) and (5,1)
35. What is the volume of a sphere, in terms of 𝜋, whose diameter is 12?
36.