Chapter 3 Section 3.1 80-87

#4 How is squaring a number reverse of finding a square root?

It's the reverse because you can find both answers even though you have 1 question.

#9 What is the area of the square with each side length?
a) 10
10 x 10 = 100 square units
b) 16
16 x 16 = 289 square units

#10 Determine the area of a square with each side length?
a) 9 squared = 81 square units
b) 11 squared = 121 square units

#15b Square root of 64 = 8

#16b Square root of 25 = 5

#18 A floor mat for gymnastics with a side length of 14 m. What is the area of the floor mat in square meters?

14 x 14 = 196  squared meters

#21 Kate is going to put a patio in her backyard. The patio stones she is using each have an area of 1 squared meters.

a) Area of the patio:  14 x 4 = 56 squared meters

b) Different dimensions with same area: 2 m by 28m

c) Is it possible it could be a square with whole side lengths  and same area? 

No, because it's not a perfect square. 

Here is a video to help you guys out. Please comment and suggest things to improve on my blog post.

Textbook Questions 2, 8, 15a, 16a, 17, & 20

2) How would you use prime factorization to determine the square root of 225?

 

 This picture is a way you can use prime factorization to determine the square root of 225.

What I did in the picture is that I made a factor tree. First I started off with the number that was given in the question that we had to find the square root. Then you find 2 factors that equal to 225. I chose 15x15. but you can use different ones if you'd like because you will still end up with the same prime factors! Next you have to find out whether 15 is a prime number. Well it's not, 3 and 5 are factors of 15. Next you figure out whether 3 and 5 are prime numbers. And they are! Now you just have to make the upside down hut and separate the prime factors equally. To prove you work you can make a diagram. Like what I did above showing how it is a perfect square.

8) Determine the prime factorization of each number. Which numbers are perfect squares?

a) 144     b)60       c)40

A)In this picture I did pretty much the same thing as I did in the first question I explained about above.

I started off at 144 then found out that 12 and 12 are the factors of 144. 12 is not a prime number so I found the factors of 12. I chose 6 and 2. Next I found that 6 isn't a prime number so you have to find the factors of 6 which is 3 and 2. Now we find out that 3 and 2 are not prime numbers so your tree stops there.

Then you separate them equally once again. This is a perfect square because it has a square root, and that you can have the same amount when you separate the number equally and multiply them. So 3x2x2=12 and 3x2x2=12. Then 12x12 = 144 or 12².  All sides must have the same length in order for it to be a square. Which it is.

B)I found the 2 factors of 60 which were 10 and 6. Then I knew that 10  and 6 are not prime numbers. So for 10 the factors are 2 and 5 and for 6 it is 3 and 2. Now you end up with prime factors at the bottom and your tree is done.

C) I found the factors of 40 and I chose 4 and 10. I knew that 10 and 4 are not prime numbers. So for 10 the factors are 2 and 5. For 4 2 and 2 are the factors. 2 and 5 are prime numbers so that is the end of your factor tree.

15)A. Evaluate the square root of 49

The square root is seven because a number that is multiplied by itself and equals that number is the square root. And if you did a factor tree the factors would be 7 and 7. 7 is a prime number so you your factor tree is done. To prove it you put 7 for one side of a square and the other factor which is seven also on the other side now think if 7 is the same as 7. It is right? So now 7 multiplied by 7 is 49 or 7².

16)A. Find the square root of 9

 The square root of 9 is 3 because in a factor tree  the factors of 9 are 3. Since 3 is a prime number the factor tree is done. To prove this I made a square diagram that show it has equal sides with the "tickies" so the sides must be the same in order for is to be a square and to be a square root of a number. So you divide the numbers evenly and you get 3 on one side and 3 on the other. Now think, is 3 the same as 3? Of coarse it is! So 3x3 or 3² gets you nine.

17) A fridge magnet has an area of 54mm ². Is 54 a perfect square? Use prime factorization to find the answer.

 54 is not a perfect square because the factors of 54 is 9 and 6. Since they are not prime numbers you have to continue. So 9's factors are 3 amd 6's factors are 2 and 3. 2 and 3 are prime numbers so thats the end. Now to find out whether it is a perfect square you have to separate the number equally so 3x2 and 3x3. Then you find out 3x2 and 3x3 is. The answers are 6 and 9. 6 is not the same as 9 so it can not be a square it is a rectangle!. So it can not be a perfect square.

20)Adam's uncle has instructions for building a shed. One page of instructions is not very clear.

 A) What is the area of a rectangle?

B) What is the length of the side of the square?

A) In the  picture the length of the rectangle was 9m and the width was 4m. So the formula to find the area is

l x w. So now you get the length which is 9 and the width which was 4 and multiply them. 9x4 = 36m².

B) Because the area of the rectangle was the 36m²  The area for the square has to be the same thing.  So instead of 9 x 4 what other factor can you use to just square and equal to 36. 6 is the answer. Here is a factor tree that I will show you, that will prove that 6 is the length of one side.

 

 

       Here is a video about square roots :)

Here is a link to help you practice! :)

Maaath

 1) Pythagorean Relationship

• The Pythagorean Relationship is the relationships between the lengths of the sides of a right triangle.
• The sum of the areas of the squares attached to the legs of a right triangle equals the area attached to the hypotenuse.

~ The hypotenuse is the longest side of a right triangle, the side of the right angle.
~ A right triangle has a right angle of 90 degrees, and this may be marked as a square.

Example:

 


2. Solve for the missing side length.







 

3. Is this a right triangle? Prove it!!!

This is not a right triangle because the legs squared do not sum up to the hypotenuse squared.

Math Textbook

Question 6:

Step 1: Find what makes 64, so 8x8=64

Step 2: Find what makes 8, 4x4=8

Step 3: Find what makes 4, 2x2,

Step 4: Circle the prime numbers

64 is a perfect square number because you can multiply 8 by it's self to make 64.

Another way of saying that is...the square root of 64 is 8, it is a whole number, so it is a perfect square.

The area: 64m². You have to remember to draw the "tickeys" to show that it is a "real" square.

11)

12)

15c)

(Blue is your question, Red is your answer.)

16c)

(Blue is your question, Red is your answer.)

19)

''

Step 1: Find the square root of 28 900

Step 2: The square root will tell what the length and width is

Step 3: Add the side together to find the perimeter which is 680

Step4: Then, the teacher told the students to do two laps, so you would multiply 680 by 2

that will give you 1360m

22)

Remember to play MangaHigh! :)

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a² + b² = c²

That formula means, that if you take the legs of the triangle, square them, then add them, you will find (c²) which is the hypotenuse². So if you have a right triangle, the legs² will equal the hypotenuse².

Angela's Scribe Post - Prime Factorization

Today in class we started off with listing the things we know about rectangles and squares.

Rectangles:

• Has four sides:
• Consisting of 2 sets of parallel sides
• The sides are labeled as "l" for length and "w" for width. (Making it different from squares)
• To find the perimeter of a rectangle you add all of the lengths and widths all together, and the formula is perimeter = 2l + 2w

Next we determined how to find the area of a rectangle:

Squares:

• 4 equal sides
• To determine the perimeter of a square you multiply its sides to four or add all of its side together. (perimeter = 4 s)

Next we determined how to find the area of a square:

After that we learned about Prime Factorization:

Prime Numbers: Have only 2 factors 1 and itself.

2, 3, 5, 7 are examples of prime number. (more here)

How to determine if something is squared:

1. Put all numbers (from the Line of Prime Factorization) onto a square:

2. Multiply each side:

If each side are equivalent to each other then the number is a squared. But in this case both side are not equivalent making the number 24 not squared.

Here's a video to help you more with squares, rectangle, and prime factorization:

Here are some games on:

Factors

Prime Factorization

Squares and Square Root

Don't forget to finish the following home works:

• Home work booklet pg. 22-25
• Play Manga High Challenges. (And other games)

Don't forget to comment!

PART II: Pythagorean Relationship

1. Answer in a short paragraph and with diagrams:

The Pythagorean relationship is the relationship between the lengths of the sides of a right triangle. The sum of the areas of the squares attached to the leg of a right triangle equals the area of the square attached to the hypotenuse.

For Example:

Legs:

• The legs are the two shorter sides of a right triangle.
• The legs of a right triangle are perpendicular to each other.

Hypotenuse:

• The longest side of a right triangle.
• Is opposite of the right angle.

2. Solve for the missing side length.

3. Is this a right triangle? Prove it!!!