Unit 12 Roots and Pythagorean Theorem
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Unit 12 Roots and Pythagorean Theorem Name: Teacher: Grade:
Transcript of Unit 12 Roots and Pythagorean Theorem
Unit 12
Roots and Pythagorean Theorem
Name:
Teacher:
Grade:
Day 1 CLASS WORK
Real Numbers
Real Number System
Vocabulary
Counting Numbers (Natural Numbers):
Whole Numbers:
Integers:
Try These:
Tell whether each is Rational or Irrational and why.
1) .424242... 2) -.5638 3) . 68
4 4) -3.312334... 5) -.000009 6)
5
Rational Real Numbers
Irrational
Rational Numbers
1) A number that can be written as a fraction where
the numerator and denominator are both integers.
2) A decimal that terminates.
3) A decimal that repeats with a pattern.
4) The square roots of perfect squares.
Irrational Numbers
1) A decimal that repeats without a pattern.
2) Pi
3) The square roots of non-perfect squares
34 7)
53454
8) 0 9) 456,812
7
10) 3.14 11) 7𝜋 12) 22
13) √ 4
81
14) √85 15) √625
Extended Response
16) a) Circle the rational number? 𝜋 √8 √81
b) Explain why the number you chose is rational.
5 √4 17) a) Circle the irrational number? √85
√16 √8
b) Explain why the number you chose is irrational.
18) Circle only the rational numbers
7.8 3.8732... 5. 7 -17
32 √361 √300 12𝜋
16 19) Which letter on the number line below best represents the value of √6?
A B C D
0 1 2 3 4 5 6
20) Place the following numbers on the number line:
3
√5 , , −0.5 , −√4 , 𝜋 2
-2 -1 0 1 2 3 4
Classwork
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
6) Which of the following is not a rational number?
1
A) 6 B) C) 7𝜋 D) 3.5 3
2) Which of the following is not a counting number
(natural number)?
A) 0 B) 1 C) 2 D) 3
7) Which of the following is a rational number?
8
A) 7.2348 … B) 𝜋 C) √25 D) 9
3) Which is an example of a whole number?
1
A) 0 B) -11 C) 2.5 D) 2
8) Which of the following is not an irrational
number?
𝜋
A) 4 B) C) √7 D) √200 3
4) Which number is an integer, a whole number, and
a counting (natural) number?
A) 0 B) -1 C) 15 D) 0.5
9) Which of the following is an irrational number?
A) 9.5 B) 3.14 C) 9𝜋 D) −0.5
5) Which of the following is a rational number?
A) 𝜋 B) −11
C) √2 D) −5.5234 …
10) Which of the following is irrational?
1
A) 0 B) 2
C) 2𝜋 D) 3.14
11) Summary: Complete the chart by putting a check in every correct column:
-1
4
5
√225
0.54 0. 3̅
5𝜋
1
0
√10
Counting
Whole
Integer
Rational
Irrational
1) Real numbers are made up of and numbers.
Rational Numbers: List examples of the following (be sure to use …)
2) Counting Numbers are .
3) Whole numbers are _ .
4) Integers are .
5) Complete the chart by putting a check in the correct column:
-4
17
0
-49
58
2,000
1
-10
25
Counting
Whole
Integer
Tell whether each description is rational or irrational:
6) Terminating decimals are .
7) Pi is .
8) Decimals that repeat without a pattern are _ .
9) Fractions are .
10) The square roots of perfect squares are .
11) The square roots of all other positive integers are .
12) Decimals that repeat with a pattern are .
13) Complete the chart by putting a check in the correct column:
-16
2
3 𝜋
2
√48
3.6
5 0. 6̅ 3. 2̅̅̅3̅
√225
Rational
Irrational
14) Change 4
5
to a decimal. Is that rational or irrational?
15) Change 22
7
to a decimal. Is that rational or irrational?
3 3
Day 2 CLASS WORK
Perfect Squares and Perfect Cubes
Perfect Squares and Perfect Cubes
Vocabulary:
Perfect square- the answer to a number multiplied by itself two times.
Square root- the number that when multiplied by itself is equal to the given number.
Perfect cube- the answer to a number multiplied by itself three times.
Cube root- a number that when raised to the third power is equal to the given number.
Radicand- the number under the radical symbol.
Part 1: Perfect Squares
List the first 15 perfect squares:
, , , , , , , , , , , , , ,
12
22
32
42
52
62
72
82
92
102
112
122
132
142
152
Examples: Simplify
1) √36 =
because ( )2 is 36 2) −√64 =
3) √. 09 =
4) √81
= 9
5) − √196
= √25
Part 2: Perfect Cubes
List the first 10 perfect cubes:
, , , , , , , , ,
13
23
33
43
53
63
73
83
93
103
Examples: Simplify
6) 3√27 = because ( )3 is 27 7) −
3√8 =
8) 3√125 =
9) √216
= 27
10) − √343
= √49
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Part 3: Equations: Solve each equation for x
Examples:
11) 𝑥2 = 81 2) 𝑥3 = 1000 3) 𝑥2 = 1.44 4) 𝑥2 = 121 5) 𝑥3 = 512
Try These:
1) √9 2) −√100 3) ±√. 0004
4) √225 25 5)
√16 √169
6) √4 + 5
7) 𝑥2 = 36 8) 𝑥2 = 1 9) 𝑥2 = 144
10) The area of a square boxing ring is 225ft
2. What is the length of one side? (a = s
2)
11) 3√125 12) −
3√64
13)
3√1
14) −3√216
3 15) √
27
√512 16)
√16
17) The volume of a cube is 125 ft3. What is the length of one side? ( v = s
3)
1
3
6) Which statement below is true?
1) What is the value of √100?
A. 4
B. 10
C. 25
D. 50
2) What is the value of √27?
A. 3
B. 5
C. 9
D. 13.5
3) Solve for x: x
2 = 256
A. x = ±16
B. x = ±15
C. x = ±16
D. x = ±128
4) Solve for y: y
3 = 216
A. y = 4
B. y = ±4
C. y = 6
D. y = ±6
5) Which statement below is true?
A. √1 = 3√1
B. √2 = 3√3
C. √4 = 3√9
D. √4 = 3√27
A. √4 = 3√4
B. √4 = 3√27
C. √16 = 3√27
D. √16 = 3√64
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 side 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 cm
3.
Part A: What is the length of one side?
Part B: What is the area of one of the faces?
25 196 1 49
36 16
9
36
* Step 1: PerfectSquare OtherFactor
Step 2: Simplify the Perfect Square
Step 3: Leave other factor in radical sign
Step 4: Write final answer
Day 3 CLASS WORK
Simplest Radical Form
Review Work: Put each in simplest radical form.
1) 2) 3) 4) 5)
6) 7) 8) 9) 10)
11) 12) 13) 14) 15)
Simplifying Radicals
Rule: Example:
Simplify
3
3
* To help with Step 1 - Perfect Square MUST be written first.
List all perfect squares up to 100 - 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
Perfect Squares:
Remember Begin with:
Examples: Put each in simplest radical form.
9 2
2
100
9 64 4
1
196
144
225
64
81
25
121
18
2
PerfectSquare OtherFactor
12 20 64
27 24 36
8 18 54
28 50 45
49 63 108
1) = 2) = 3) =
4) 5 = 5) 3 = 6) 2 =
Try These: Put each in simplest radical form.
1) = 2) 4 = 3) 8 =
4) = 5) 2 = 6) 10 =
7) 2 = 8) 5 = 9) =
24 40 8
99 28 64
12 32 18
45 50 27
98 40 9
Put each in simplest radical form.
1) = 2) = 3) 5 =
4) 4 = 5) 2 = 6) =
7) 5 = 8) = 9) =
10) = 11) 3 = 12) =
13) 3 = 14) 5 = 15) 4 =
27
75 54
50
12 121
48
125
20
72 8
List the first 10 Perfect Squares:
Place all radicals in Simplest Radical Form
1) 2) 4
3) 4) 3
5) 6) 7
7) 8) 8
9) 10) 4
11) 12) 3 175
13) 14) 5
15) 16) 3
80
40
24
108
63
144
32
225
200
18
28 44
1000
1,000,000 200
128
27
98
List the first 10 Perfect Squares:
17) 18) 2
19) 20) 4
21) 22) 6
23) 24)
25) 26)
27) 5 28)
29) 3 30) 5
31) 2 32)
33) 34) 3
35) 3 36) 6
37) 8 38)
39) 40) 2
45
50
300
250
84
169
60
90
52
500
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Day 5 CLASS WORK
Rounding
Rounding non-perfect square roots, non-perfect cube roots, and equation answers
Part 1: Rounding
1, 2 3 4 . 5 6 7 8
Steps to Rounding: 1) Underline the number that you are rounding to.
2) Draw an arrow to the number after it.
3) 4 and lower stay the same OR 5 and higher go up.
Examples:
1) Round the following to the nearest whole number: 2) Round the following to the nearest tenth:
8132 = 8.9437 =
3) Round the following to the nearest hundredths: 4) Round the following to the nearest thousandths:
3987.42915 = 9534.0934 =
Try These:
Round the following to the nearest whole number:
1) 159.513 2) 0.783 3) √55
Round the following to the nearest tenths:
4) 2,374.928 5) 12.63 6) √87
Round the following to the nearest hundredths:
13) 35.689 16) 0.95238 18)
√320
Box the number.
754.
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Part 2: Non-Perfect Square Roots, Non-Perfect Cube Roots, and Equation Answers
Vocabulary:
Non-Perfect square-
Non-Perfect cube-
Examples:
1) Round √3 to the nearest tenth 2) Approximate −√27 to the nearest tenth.
3) Round ±3√20 to the nearest hundredth. 4) Approximate
3√638
to the nearest tenth.
5) Round the answer 𝑥2 = 30 to the nearest tenth. 6) Round the answer 𝑥3 = 30 to the nearest tenth.
Try These:
Round to the nearest tenth
1) √8 2) √108 3) −√6 4) 𝑝2 = 140 5) √80
9
6) 3√12 7) −
3√60 8)
3√100 9)
3√80 10) −
3√75
Approximate to the nearest whole #.
11) 𝑥2 = 30 12) √300 13) − √195
√7
14) √
220
20
15) √1 + 7
16) x3
= 125
3
17) 𝑝3 = 21 18) √175
25
19) 𝑥3 = 1030 20) 𝑥3 = 21
20) The area of a square boxing ring is 22ft2. What is the length of one side to the nearest
hundredth? (a = s2)
21) The volume of a cube is 240 ft
3. What is the length of one side to the nearest hundredth?
( v = s3)
1) What is the value of √16?
6) Which of the following has the correct
approximation to the nearest whole number?
A. 3√200 = 6
B. 3√7 = 1
C. 3√86 = 5
D. 3√52 = 3
A. 4
B. 10
C. 25
D. 50
2) What is the value of 3√37 to the nearest 7) Mateo thinks the cube root of 120 is approximately
5. Is he correct? Explain how you know. whole number?
A. 4
B. 3
C. 2
D. 3.3
3 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.
3) Find the radicand: √ = 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
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?
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
To estimate the square root of a non-perfect square without a calculator:
Rule:
1. Find the closest perfect square less than the number 2. Find the closest perfect square greater than the number
3. The square root of the non-perfect square must be between the square root of these two perfect squares
3
Day 6 CLASS WORK
Compare and Order Real Numbers
Estimating, Comparing and Ordering Square Roots
Review: List the first 15 perfect squares
, , , , , , , , , , , , , , 12
2
2 3
2 4
2 5
2 6
2 7
2 8
2 9
2 10
2 11
2 12
2 13
2 14
2 15
2
Vocabulary:
Integers- whole numbers and their opposites (Ex: {….-3, -2, -1, 0, 1, 2, 3,…}
Consecutive- one number right after another.
Inequality Symbols-
< > ≤ ≥
Less Than Greater Than Less than or equal to Greater than or equal to
Examples:
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) √27 √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 , 3√125
0 1 2 3 4 5
3
3
Try These:
Find the two consecutive integers the square root lies between.
1) √29 2) √110 3) −√15 4) −√72
Compare: Use >, <, ≤, ≥, or =
5) 16 √16 6) √121 11 7) -16 -√225 8) √8 √45
Order from least to greatest
9) √2 , 5.85, 42, − √33 10) √125 ,
√8 , −√100 , -√201
11) Which of the following is the best estimate for √58 ?
A) 2.9 B) 7.7 C) 7.6 D) 5.8
12) Place the following numbers on the number line:
√2 , √18 , 0.5 , √4 , √11
0 1 2 3 4 5
3
3
3
Find the two consecutive integers the square root lies between.
1) √19 2) √99 3) √226 4) √42 5) −√6 6) −√55
Compare: Use >, <, ≤, ≥, or =
7) 7 √7 8) √121 11 9) -16 -√225 10) √8 √45 11) 3 √9
Order from least to greatest
12) √125 , √8 , √100 , -√201 13) √224, 103 √57 , 5.89
14) Place the following numbers on the number line:
√9 , √5 , 2 , 1.2 , √24 , 3√125
0 1 2 3 4 5
Solve:
15) √169 16) √27 17) 𝑥2 = 144 18) 𝑝3 = 125
Approximate the value of each to the nearest whole number.
19) √8 + 𝜋 20) √120 - √18 21) 4 × √38 22) 3√47
7.
The Pythagorean Theorem:
A. only works in triangles.
B. Sides a and b are called .
C. Side c is called the .
*It is the side of the triangle.
Day 7 CLASS WORK
Pythagorean Theorem
State the side length that would represent the hypotenuse of the following triangles:
1. 3, 4, 5 2. 5, 12, 13 3. 8, 17, 15 4. 39, 15, 36
5. A. Label each side in the picture as a leg or the hypotenuse.
B. There is another hypotenuse in the picture. What is its name?
Find the missing sides of the right triangles:
6.
9. What is the length of the hypotenuse of a right triangle that has one leg with a length of 9 inches and the
other leg with a length of 12 inches?
10. If the length of the hypotenuse of a right triangle is 10 feet and one leg measures 6 feet, what is the measure
of the other leg?
8.
1.
Find the missing side of the following right triangles:
5. The distance across a pond cannot be directly measured. A land surveyor takes some other measurements
and uses them to find d, the distance across the pond.
What is the distance across the pond?
2.
3. 4.
Day 8 CLASS WORK
More Practice with Pythagorean Theorem
1. 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐 2. 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐 𝟑𝟐 + 𝟒𝟐 = 𝒙𝟐 𝟑𝟐 + 𝒙𝟐 = 𝟒𝟐
𝟗 + 𝟏𝟔 = 𝒙𝟐 𝟗 + 𝒙𝟐 = 𝟏𝟔
√𝟐𝟓 = √𝒙𝟐 −𝟗 − 𝟗
5 = x √𝒙𝟐 = √𝟕
𝒙 = √𝟕
Recall:
A. Simplify all square roots.
B. When multiplying square roots, multiply the coefficients, multiply the radicands, and simplify.
Examples
For #1 and 2, Round to the Nearest Tenth.
1. 2.
For #3 -9, Find the length of the missing side. Leave answers in Simplest Radical Form.
3. 4. 5.
Find the length of the missing side. Leave answers in simplest radical form.
6. a = √2 and b = √2
7. b = 4 and c = 4√5
Decide whether each of the following are right triangles:
1. 9, 40, 41 2. 7, 12, 18 3. 8, 17, 15
4. 𝟏, 𝟐, √𝟓 5. √𝟐, 𝟑, √𝟏𝟏 6. √𝟐, √𝟑, √𝟓
Find the length of the missing side in Simplest Radical Form. Then, round to the nearest tenth.
1. 2. 3.
4. a = 8 and b = 8
5. a = 2√2 and b = 2√3
Determine whether the following are the sides of a right triangle.
6. 8, 6, 10
7. 7, 25, 24
8. 14, 19, 23
Drawing with Labels
Day 9 Classwork
Word Problems with Pythagorean Theorem
For each of the following, first draw the picture and then solve for the missing dimension:
1) In a square, the sides equal 8cm. What is the length of the diagonal in simplest radical form?
2) You need to climb to the roof of your house using a ladder. Your house is 30 feet high and you start with the ladder 14
feet away from your house. How long is the ladder to the nearest tenth?
3) The length of a base of a sail is 13 meters and a height of 14 meters. How long is it from the end of the base to the top
of the sail to the nearest hundredth?
4) A baseball diamond is actually a square with 90 feet between bases. What is the distance between home plate and
second base to the nearest ten-thousandth?
5) A suitcase measures 24 inches long and 18 inches high. What is the diagonal length of the suitcase?
Drawing with Labels
Drawing with Labels
Solve For Missing Side with Work Shown Drawing with Labels
Solve For Missing Side with Work Shown Drawing with Labels
Solve For Missing Side with Work Shown
Solve For Missing Side with Work Shown
Solve For Missing Side with Work Shown
6) Mark rides his bike 12 miles North and then 6 miles east. What is the distance, in simplest radical form, back to his
starting point?
7) A person who is 6 feet tall casts a shadow that is 5 feet long. What is the distance between the head of the person
and its shadow to the nearest tenth?
8) Oscar’s dog house is shaped like a tent. The slanted sides are both 5 feet long and the bottom of the house is 6 feet
across. What is the height of his dog house at its tallest point?
9) In a computer catalog, a computer monitor is listed as being 20cm. The distance is the diagonal distance across the
screen. If the screen measures 10cm in height, what is the actual width of the screen in simplest radical form?
