8th Grade
Unit 5 – Irrational
Numbers Using Geometry:
Packet 1 of 4
Mr. Rothdiener
Name: _____________________________________
Packet Outline, Page 1
 5.1: Square Roots (8.EE.A.2), Pages 2-4
o I can calculate the square root of perfect squares.
o I can approximate the location of square roots of whole numbers on the
number line.
 5.2: Simplify Square Roots (8.EE.A.2), Pages 5-7
o I can use factors of a number to simplify square roots.
 5.3: The Pythagorean Theorem (8.G.B.7), Pages 8-11
o I can use the Pythagorean Theorem to determine missing side lengths
of right triangles.
 5.4: The Pythagorean Theorem and its Converse (8.G.B.7), Pages 12-14
o I can use the Pythagorean Theorem to determine missing side lengths
of right triangles.
o I can illuminate the converse of the Pythagorean Theorem through
computation of examples and counterexamples.
 5.5: Distance on the Coordinate Plane (8.G.B.8), Pages 15-19
o I can determine the distance between two points on a coordinate plane
using the Pythagorean Theorem.
 5.6: Applications of the Pythagorean Theorem (8.G.B.7), Pages 20-23
o I can use the Pythagorean Theorem to determine missing side lengths of right
triangles formed from real-world situations.
 Quiz 5.1 Review (8.EE.A.2, 8.G.B.7, 8.G.B.8), Pages 24-25
o I can simplify both perfect and non-perfect square roots.
o I can understand and apply the Pythagorean Theorem.
1
5.1: Square Roots (8.EE.A.2)
Bell Work:
Complete the following calculations:
12 =
62 =
22 =
72 =
32 =
82 =
42 =
92 =
52 =
102 =
Whole-Class:
1.) Determine the positive square root of 81, if it exists. Explain.
2.) Determine the positive square root of 225, if it exists. Explain.
3.) Determine the positive square root of −36, if it exists. Explain.
4.) Determine the positive square root of 49, if it exists. Explain.
2
Guided Practice:
5.) Place the numbers √1, √4, √9, and √16 on the number line.
6.) Place the numbers √2 and √3 on the number line.
7.) Place the numbers √5, √6, √7, and √8 on the number line.
8.) Place the numbers √10, √11, √12, √13, √14, and √15 on the number line.
Independent Practice:
Determine the positive square root of the number given. If the number is not a
perfect square, determine which whole number the square root would be closest to,
and then use guess and check to give an approximate answer to one or two decimal
places.
9.) √49
10.) √62
11.) √122
12.) √400
13.) Which of the numbers in Exercises 9-12 are not perfect squares? Explain.
3
HOMEWORK 5.1
Determine the positive square root of the number given. If the number is not a
perfect square, determine the integer to which the square root would be closest.
1.) √169
2.) √256
3.) √81
4.) √147
5.) √8
6.) Which of the numbers in Problems 1–5 are not perfect squares? Explain.
7.) Place the following list of numbers in their approximate locations on a number
line.
√32, √12, √27, √18, √23, and √50
4
5.2: Simplify Square Roots (8.EE.A.2)
Bell Work:
Without using a calculator, find the square root of the following.
25
0
.49
 16
Whole-Class:
1.) 54
2.) 28
3.) 90
4.) 98
5.) 108
6.) 32
7.) 300
8.) 72
5
Guided Practice:
1
48
2
11.)  5 44
12.) 6 24
14.)  10 52
15.) 2 99
16.)  72
18.) 18
19.) 20
20.) 63
9.) 4 50
10.)
13.) 9 40
Independent Practice:
17.) 50
6
HOMEWORK 5.2
Simplify the square roots as much as possible.
1.) √24
2.) √72
3.) √128
4.) √48
7
5.3: The Pythagorean Theorem (8.G.B.7)
Bell Work:
Finish the pattern to 202
12 = 1
22 = 4
32 = 9
Whole-Class:
Find the length of the third side.
1.) a = 3; b = 4
2.) a = 5; b = 12
3.) b = 15; c = 17
4.) a = 16; c = 20
8
Express the length of the third side to the nearest hundredth.
5.) a = 3, b = 3
6.) a = 4, b = 8
Guided Practice:
Find the missing side in:
a) exact value.
b) rounded to the nearest tenth.
7.)
8.)
6
4
x
9
Independent Practice:
9.) Given a rectangle with dimensions 5 cm and 10 cm, as shown, find the length of
the diagonal.
10.) Given a rectangle with side lengths of 8 cm and 2 cm, as shown, what is the
length of the diagonal?
10
HOMEWORK 5.3
1-4: Use the Pythagorean Theorem to determine the unknown length of the right
triangle (exact value).
1.)
2.)
3.)
4.)
11
5.4: The Pythagorean Theorem and its Converse (8.G.B.7)
Bell Work:
Is the triangle a right triangle?
Whole-Class:
Is the triangle a right triangle?
1.) 3, 2, 4
2.) 25, 7, 24
3.) 12, 8, 5
Guided Practice:
Solve for x. Express your answer rounded to the nearest tenth, if necessary.
4.)
5.)
x
6.)
2x
4
12
2x
6
7
10
12
x
12
7.) x
8.)
8
x
13
9.)
6
18
x
14
Independent Practice:
The numbers in the diagrams below indicate the units of length of each side of the
triangles. Is each triangle shown below a right triangle?
10.)
11.)
13
HOMEWORK 5.4
1-3: Is the triangle a right triangle?
1.) 5, 4, 3
2.) 11, 12, 15
3.) 13, 5, 12
4-5: Find the value of x in each right triangle. Give your answers rounded to the
nearest hundredth if necessary.
4.)
2
5.)
x
1
√13
√5
x
14
5.5: Distance on the Coordinate Plane (8.G.B.8)
Bell Work:
What is the distance between the two points 𝐴 and 𝐵 on the coordinate plane?
What is the distance between the two points 𝐴 and 𝐵 on the coordinate plane?
15
Whole-Class:
1.) What is the distance between the two points 𝐴 and 𝐵 on the coordinate plane?
Round your answer to the tenths place.
2.) Given two points 𝐴 and 𝐵 on the coordinate plane, determine the distance
between them. First, make an estimate; then, try to find a more precise answer.
Round your answer to the tenths place.
16
Guided Practice:
3-6: Determine the distance between points 𝐴 and 𝐵 on the coordinate plane.
Round your answer to the tenths place.
3.)
4.)
17
Independent Practice:
5.)
6.)
18
HOMEWORK 5.5
Determine the distance between points 𝐴 and 𝐵 on the coordinate plane. Round
your answer to the tenths place.
1.)
2.)
19
5.6: Applications of the Pythagorean Theorem (8.G.B.7)
Bell Work:
Determine the length of the hypotenuse of the right triangle shown if a = 6 and b = 13.
Whole-Class:
1.) Suppose you have a ladder of length 13 feet. To make it sturdy enough to climb
you must place the ladder exactly 5 feet from the wall of a building. You need to
post a banner on the building 10 feet above the ground. Is the ladder long
enough for you to reach the location you need to post the banner?
20
2.) Emma is in a hot air balloon that has just taken off and is now floating 20 meters
above its launching point. Doug is standing on the ground, 8 meters away from
the launching point. How far apart are Emma and Doug to the nearest tenth of a
foot?
Guided Practice:
3.) The mast of a fishing boat is supported by a sturdy rope that extends from the
top of the mast to the deck. If the mast is 20 feet tall and the rope attaches to the
deck 10 feet away from the base of the mast, how long is the rope, to the nearest
tenth of a foot?
4.) Clara is fishing from a small boat. A fish swimming at the same depth as the
hook at the end of her fishing line is 2√5 feet away from the hook. If Clara is
6√3 feet away from the fish, how far below Clara is the hook to the nearest
foot?
21
Independent Practice:
5.) You have a 15-foot ladder and need to reach exactly 9 feet up the wall. How far
away from the wall should you place the ladder so that you can reach your
desired location?
6.) A ladder is placed 5 feet from the foot of a wall. The top of the ladder reaches a
point 12 feet above the ground. Find the length of the ladder to the nearest foot.
22
HOMEWORK 5.6
1.) A 20-foot ladder is placed 12 feet from the wall, as shown. How high up the
wall will the ladder reach?
2.) A rectangle has dimensions 6 in by 12 in. What is the length of the diagonal of
the rectangle?
3.) A ladder is against a wall. From the base of the wall to the top of the ladder is 90
in. The base of the ladder is 30in from the wall. What is the length of the ladder
to the nearest inch?
23
Quiz 5.1 Review
(8.EE.A.2, 8.G.B.7, 8.G.B.8)
Use your knowledge of math to complete each of the following questions. Show
your work!
1-5: Determine the positive square root of the number given.
1.) √81
2.) √25
3.) √144
4.) √169
5.) √9
6-7: Simplify the square roots as much as possible.
7.) √98
6.) √54
8-9: Use the Pythagorean Theorem to determine the unknown length of the right
triangle.
8.)
9.)
24
10-11: Is the triangle a right triangle?
10.) 10, 8, 6
11.) 13, 14, 18
12.) Determine the distance between points 𝐴 and 𝐵 on the coordinate plane. Round
your answer to the tenths place.
13.) The top of a ladder is placed against a wall 20 feet above the ground. The
length of the ladder is 25 feet. How far from the base of the wall should the
bottom of the ladder be placed?
25