Grade 8 Math
Unit 1
Square Roots and the
Pythagorean Theorem
Section 1.1
Square Numbers and Area Models
Page 6
When we multiply a number by itself we are squaring the number
The square of 3 is 9 since 3 × 3 = 9
We say, 3 squared is 9.
9 is called a square number or perfect square.
Common Perfect Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169 etc.
Do page 8, #4 together
Example:
Find the area of a square with side length 6 units.
6 units × 6 units = 36 units2
So the area is 36 units2.
Try page 8, #5
Example:
Use a diagram to show the number 25 is a square number.
Try page 9, #10
Example:
Find the side length of a square with area 49 units2.
Since 7 × 7 = 49, the length of the side must be 7 units.
Find the side length of a square with area 16 units2.
Since 4 × 4 = 16, the length of the side must be 4 units.
Try page 9, #11
Example:
Which number is a square number, 12 or 64?
12 is not a square number since it is between the square numbers 9 and 16.
3 × 3 = 9 and 4 × 4 = 16
64 is a square number since 8 × 8 = 64
Try page 9, #12
Example:
The floor of a room has area 100 m2.
a.) What is the length of one side of the room?
Since 10 × 10 is 100, the side length is 10 m.
b.) How much edging is needed to go around the room?
Each side is 10 m. There are 4 sides.
4 × 10 m = 40 m of edging (door included)
Try page 9, #16 ab
Section 1.2
Squares and Square Roots
Page 11
If a number can be drawn as a square on grid paper we say it is a perfect square. The length of
the side is the square root of the number.
Ex.
Find the square root of 64.
Since the side length of 64 is 8, the square root of 64 is 8. This is written as:
The symbol
Example:
64 = 8
means square root.
Find the square of 7.
The square of 7 is: 72 = 7 × 7 = 49
The square of 7 is 49.
Find the square of 8.
82 = 8 × 8 = 64
The square of 8 is 64.
Note: An exponent of 2 just means we
multiply a number by itself.
Ex: 52 = 5 × 5
Try page 15, #5,6
Example:
Find
36 .
Since 62 = 6 × 6 = 36, then
Note:
36 = 6
Try page 15, #7,8
1= 1
4=2
9 =3
16 = 4
25 = 5
36 = 6 etc.
Example:
Use a table. List the factors of 40 and 64. Which numbers are square numbers?
40
1
2
4
5
64
40
20
10
8
1
2
4
8
64
32
16
8
40 is not a perfect square since none of the factors occur twice in the same row.
64 is a perfect square since the 8 occurs twice in the same row.
Try page 15, #9, 10, 11,13
Example:
Find the square root of 52 and 72.
52
=
= 5
25
72
=
= 7
49
Try page 16, #14, 15, 19
Section 1.3
Measuring Line Segments
Page 20
Complete page 20, #3, 4
Example:
Given the area of a square, find the side length
A = 64 cm2
Well,
64 = 8
so the side length is 8 cm.
Given the side length of a square, find the area.
s = 5 cm
Well, 52 = 25 cm2
so the Area is 25 cm2
Example:
Find the area and side length of the black square.
Try page 20, #5, 6
To find the black square’s area we can find the total area of the grid and subtract the area of
the 4 outer triangles.
Total Area of Grid: = length × width
Area of Outer Triangle =
bh
2
= 8 units × 8 units
=
(2)(6)
2
= 64 units2
=
12
2
= 6 units2
There are 4 triangle so, Total Area of Triangles = 4 × 6 = 24 units2
So, Area of Black Square = 64 uints2 – 24 units2
= 40 units2
If the area of the black square is 40 units2 then the side length =
Since 40 is not a perfect square we can leave our answer as
estimate the value of
Try page 20, #8,9
40 as a decimal.
40 .
40 . In section 1.4 we will
Section 1.4
Estimating Square Roots
Notice the pattern:
99
=9
5 5
=5
13 13 = 13
16  16 =16
Page 25
We can check to see if the
answers are correct.
99 =
81 = 9
5 5 =
25 = 5
The answer is the number under the square root sign (Both numbers under the root sign must
be the same).
Try page 25, #4
Example:
a.)
Between which two consecutive whole numbers is each square root?
6
4 = 2
6
Choose the closest perfect squares
below and above 6 and find their
9 = 3
square roots.
The answer is between 2 and 3.
Answer: 2 and 3
b.) 13
9 = 3
13
16 = 4
Choose the closest perfect squares
below and above 13 and find their
square roots.
The answer is between 3 and 4.
Answer: 3 and 4
Try page 25, #5 a, b, c
Example:
Use the number line. Put the square root on the line to show its estimated value.
a.) 10
a.)
b.)
b.)
c.)
41
d.) 56
31
10  3.1
Since 10 is slightly more than 9,
the answer should be just above
3. Estimate 3.1
16 = 4
Note:  means approximately
9 = 3
36 = 6
c.)
25 = 5
d.)
49 = 7
41  6.4
31  5.6
56  7.4
49 = 7
36 = 6
64 = 8
Answer:
3
3.5
4
4.5
10
5
5.5
6
31
6.5
41
7
7.5
8
56
Try page 25, #8
Example:
Is the whole number less than, equal to or greater than the given square root?
a.) 5 or 10 Well
9 = 3 so 10 is slightly more than 3. Therefore 5 is bigger than 10
b.) 9 or
81 = 9 so
84 Well
Try page 25, #9 a,b,c
84 is slightly more than 9.
Therefore 9 is smaller than
84
Example:
Which whole number is
49
53 closest to?
Choose the perfect square closest to 53 and
take its square root.
=7
Perfect Square closest to 53 is 49.
The answer is 7.
Try page 25, #10
Example:
What is the approximate side length of a square with area 56 cm 2. Give the
answer to one decimal place. Use a calculator.
Answer:
The side length is the square root of the area.
56 = 7.48 cm2
Try page 26, #13
The answer rounded to 1 decimal place is 7.5 cm2.
Section 1.5
The Pythagorean Theorem
Page 31
Note: The hypotenuse is the
largest side of a right
triangle. It is opposite the 90o
angle.
Note: 25 – 16 = 9
25 - 9 = 16
16 + 9 = 25
Note:
The area of the small squares added together will always equal the area of the
large square on the hypotenuse of a right triangle.
This is a relationship known as the PYTHAGOREAN THEOREM.
Try page 34, #3, 4
To find the length of the hypotenuse we use the
formula a2 + b2 = c2. This formula is called the
Pythagorean theorem. Sides a and b refer to the legs
of the triangle. Side c always refers to the hypotenuse.
Example:
Find the length of the hypotenuse.
a2 + b2 = c2
32 + 62 = c2
9 + 36 = c2
45
= c2
45
= c2
6.7
= c
The length of the hypotenuse is 6.7 cm.
Try page 34, #5 a, b, c
We can use a2 + b2 = c2 to find the missing lengths of
the legs of a right triangle when we are given the
length of the hypotenuse.
Example:
Find the length of the missing side.
Note: It does not matter
which leg is named a or b.
a2 + b2 = c2
152 + b2 = 202
225 + b2 = 400
225 - 225 + b2 = 400 - 225
b2 = 175
b
=
b
= 13.2
175
To solve the equation, b
must be by itself. Therefore
subtract 225 from both
sides of the equation.
Try page 34, #6 a, b, c
Example:
Find the length of the diagonal. Give your answer to 2 decimal places.
a2 + b2
= c2
52 + 122 = c2
25 + 144 = c2
Note: In a rectangle,
opposite sides are equal.
169 = c2
169 = c
13 =
The length of the diagonal is 13 m.
c
Try page 35, #9, 13c
Section 1.6
Exploring the Pythagorean Theorem
Page 39
Try page 43, #3 together
Example:
Is the triangle a right triangle?
If the set of 3 whole numbers given
in the diagram satisfy the conditions
of the Pythagorean Theorem, the
triangle is a right triangle.
If both sides of the equation are equal, we have a right triangle.
a2 + b2
= c2
122 + 52
= 132
144 + 25
= 169
169
= 169 The triangle is right ∆
Note: A set of 3 such as the above (5, 12, 13) are known as a
Pythagorean Triple since they satisfy the conditions of the Theorem.
If 122 + 52 ≠ 132 then the numbers 5, 12, 13 would not be a
Pythagorean Triple.
Another example of a Pythagorean Triple is 3, 4, 5 since
32 + 4 2 = 52
Note: The largest of the 3 numbers in a Pythagorean Triple refers to the
hypotenuse of the triangle.
Try page 43, # 4ab, 6a,b,c,d, 8, 9, 11a,12ab
Section 1.7
Example:
Applying the Pythagorean Theorem
Page 46
John is laying a foundation for a garage with dimensions 10 m by 6 m. To check
that the foundation is square, John measures a diagonal. How long should the
diagonal be? Give your answer to one decimal place.
a2 + b2
= c2
102 + 62
= c2
100 + 60 = c2
160 = c2
160 = c
12.6 = c
The diagonal should be 12.6 m.
Try page 49, #4a, 5a, 6, 8a, 9, 10, 13, 16