Grade 8 - Unit One

Square Roots and The Pythagorean Theorem Name: __________________

Booklet1−1.1 : Square Numbers∧AreaModels

In this section we will be looking at the relationship between the area of a square and its side length.

Square Side Length Area

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So the area of the square is found using the formula:

A = S x S orA =S2

If we can represent an area using squares then

it is a perfect square or square number. For example,

the numbers 1, 4 and 9 are all perfect squares.

Finish the table on the right. You will need to remember

these prefect squares:

Do you think that 20 is a perfect square? Explain:

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There are 4 ways to determine if a number is a perfect square:

1. Try to draw the square2. Write a division sentence to show that the quotient is equal to the divisor3. Find the factors of the number4. Prime factorization

We will now look at how to use each of these criteria:

1) Try to Draw the Square

Is 36 a perfect square?

Is 20 a perfect square?

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Conclusion

2) Write a division sentence to show that the quotient is equal to the divisor

Is 36 a perfect square?

If you can write a division sentence so that the quotient is equal to the divisor:

36 ÷ 6 = 6

Is 20 a perfect square?

Conclusion:

3) Find the factors of the number

Ex. 36 - 1, 2, 3, 4, 6, 9, 12, 18, 36

Note: When a number has an odd number of factors it is a square number.

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Ex: 20 is not a perfect square since:

 20 - 1, 2, 4, 5, 10, 20 (even number)

Conclusion:

4) Prime Factorization

Prime Factors - only has one and itself as its factors.

Prime Factorization - is the prime numbers that multiply to give you the original number ( Bottom row of a factor tree). If each number has a pair, then it is a perfect square:

Use a factor tree to decide if 36 is a perfect square:

Use a factor tree to decide if 20 is a perfect square:

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Conclusion:

Practice:

Use these 4 criteria to show that 16 is a perfect square:

Use these 4 criteria to show that 28 is not a perfect square.

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1.2 :Squares∧Square Roots

Complete the table on the right:

Let’s look at the difference between a “square” and a “square root”:

Square Square RootDefinition

Multiply number by itself.What number, multiplied by itself, make the number under the symbol.

Symbol 42=4×4=16 √64=8 ,since8×8=64

Complete the following questions:

1) Square the following:7

To complete on loose leaf: Pages 9 # 8-12

a) 9 b) 3 c) 1 d) 23 e) 16

2) Find each square root:

a)√9 b) √64 c) √49 d) √1 e) √484

There are various techniques to find the square root of a perfect square:

1) By understanding the definition of “square root” and remembering the following:

2) Using factors to find the square root:

To find √36 , list all the factors from least to greatest:

Since the middle number doesn’t have a partner, it must multiply with itself so √36=6.

Conclusion:

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3) Using Prime factorization to find a square root:

To find √36 ,make a factor tree:

We have a pair of 2’s and a pair of 3’s:

2×3=6 , so√36=6

Conclusion:

1.4 :Estimating SquareRoots−wewill return¿1.3

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To complete on loose leaf: p. 15 & 16 # 5 to 8, 10, 11, 13 to 15

We will use number lines to help us estimate the square root of non-perfect squares:

If you have not memorized this, now is the time!!!

What is √14 ?

Since 14 is not a perfect square we must estimate. Between what two perfect squares does 14 fall between?

14 falls between 9 and 16, so √14 falls between √9 and √16 or 3 and 4. So

√14 ~ 3.7

√9 √16

3 3.7 4

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1) Estimate each square root. SHOW WORKINGS!!

a) √55 b) √100

c) √37 d)√62

e) √136 f)√4×4

1.5 :The PythagoreanTheorem

Pythagorean Theorem

Recall the Pythagorean Theorem:

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To complete on loose leaf: pages 25 & 26 # 4 to 6, 8 to 11, 14

a

b

ca2

b2

c2

a2+b2=c2

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Booklet1−1.6 :Exploring the PythagoreanTheorem

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To complete on loose leaf: 34 & 35 # 4 to 6, 9 & 13

Determine if the three sides given form a right triangle. If not, is it an acute or obtuse triangle?

1) 6 cm,8 cm, 10 cm  2) 5 cm, 13 cm, 9 cm

3) 5 cm, 4 cm, 6 cm

Ex. 1) Is 7, 24, 25 a Pythagorean triple?

 2) Is 12, 13, 15 a Pythagorean triple?

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Booklet 1−1.7 : Applying the PythagoreanTheorem

Please note that this will be completed through hands on activities in the library.

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To complete on loose leaf 43 & 44, # 3 to 7