chapter 1: prior skills & knowledge

September 02, 2014

and Surface Area

Chapter 1

Chapter 1: Prior Skills & Knowledge

Squares & Square Roots

- Terminology- Whole number squares- Approximation- Pythagorean Theorem

September 02, 2014


Terminology: Ex. 5 x 5 = 25

The square of 5 is 25The square root of 25 is 5

Try These:

What is the square root of 16?

What is the square of 9?

Which number has a square root of 10?


Whole number squares When a number has a whole number square root it is said to be a perfect square. You should be familiar with all perfect squares up to 202 = 400

Try These:

Which of the following is not a perfect square?

9, 25, 49, 64, 72, 81

A square has an area of 64 cm2. What is it's side length?

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Approximation If you know your perfect squares, you can easily estimate non-perfect squares.

Ex: What is 26

Try This:

Which whole number is closes to 55?


Pythagorean Theorem a2 + b2 = c2

Where 'C' is the hypotenuse

Try This:

Label the a, b, and c sides of this triangle:

What is the value of c?

5

12

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Pythagorean Theorem What about when you are trying to find the value of 'a' or 'b'?

Ex. 20

15

?

Now you try:14

10

?

This square playground has an

area of 400 m2 and it needs a fence. What length of fence is required to surround the playground?

Chapter 1 Skills review....

September 02, 2014

Chapter 1 Skills review....What is the side length of this square? What is the area?

= 1 cm2

Is the answer 3.6 or 13 ? Is there a difference

Chapter 1 Skills review....

What is a perfect square?

Are the following numbers perfect squares?

Square Square Root Perfect Square (y/n)

64

15

12.96

29

136.89

324

0.04

Yes

No

Yes

No

Yes

Yes

Yes

September 02, 2014

1.1 Square Roots of Perfect Squares

A square root is a number which when multiplied by itself, results in another number (called a square)

Ex. 5 x 5 = 25 √25 = 5Without using your calculator, make a list in your notebook of the squares of all whole numbers between 1 and 20:

Square Root Square

112439

A = 81 m2

The square of the side length of a square give us the area:

A = bh or A = s2

What is the area of a square with a side length of 11 cm?

What is the side length of a square with an area of 11 cm2

Calculator Check! Know how to use square root and square buttons! What about with a fraction?

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A square can be either perfect or non-perfect. A square is perfect if it's square root is a rational number.*

*Rational numbers include integers, fractions, and all decimal numbers that either "end" or have repeating sequences

Square:

Side Length/Square Root:

Perfect (Y/N):

4 cm2 25 cm2 10.24 cm21296 cm2

25 27 cm25 cm2

1.1: Square Roots of Perfect Squares (Fractions)

Let's review what we know about squares & square roots as well as what we know about fractions:

a) We know that the side length of a square is equal to the square root of the area:

ex:

b) We also know that we can represent fractions in the following way:

September 02, 2014


So what if we combined those two concepts together?

How could we represent the side lengths of the red square as fractions?

What is the total area of the red square written as a fraction? In other words, how many boxes are shaded red out of the total?

What is the relationship between the side length of the square, and it's area?


Try one on your own!

In the previous example, the area was . In this example it is . Does anything stand out about

the numerator & denominator in each? Come to class prepared to discuss your findings.

49100

925

What is the side length of the red square, written as a fraction?

What is the total area of the red square, written as a fraction?

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A fraction in simplest form is a PERFECT SQUARE if it can be written as a product of 2 equal fractions.

Also, if the numerator is a perfect square AND the denominator is a perfect square, then the entire fraction is a perfect square! (Also called rational numbers)

79

x7

9 =4981

4981

is therefore a perfect square

50128

225139

927

Which of the following are perfect squares?

Working with decimals:

- When you square a decimal, the product is smaller:

- Decimal numbers follow the same patterns as non-decimals:

0.5 x 0.5 = 0.49

0.8 x 0.8 = 0.64

Is it possible for a square to be smaller than it's square root?

3 x 3 = 9 0.3 x 0.3 = 0.09 0.03 x 0.03 x 0.009 300 x 300 = 90000

√9 = 3 √0.09 = 0.3 √0.009 = 0.03 √90000 = 300

September 02, 2014

1 2 3 4 5 6 7 8 9 100

Arrange the square roots on the number line without using your calculator:

√36√ 0.16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

√ 80

1.2 Square Roots of Non-Perfect Squares

Benchmarking:

September 02, 2014

1.2 Square Roots of Non-Perfect Squares

5 m

1.5 m

A 5 m ladder is placed against a wall. The base of the ladder is 1.5 m from the wall. How far up the wall does the ladder reach?

What happens when you try to find the square root of 5.5 on your calculator?

√5.5 = ?

To find the square root of a non-perfect square, we can benchmarks & estimates, or use a calculator!

Benchmarking & Estimating Calculator

√27 = ? √= ?10

26√0.38 = ?

√= ?31

8

September 02, 2014

The Pythagorean Theorem often involves non-perfect squares:

a2 + b2 = c2

a

bc

18 cm

14 cm

h

Steps:1. Identify the hypotenuse (c2)2. Replace known variables3. Solve for the unknown using your mad square root skills!

Find the value of the unknown side

2 m

h

x 4 mm

2.7 mm

14 cm

x

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Review: Surface Area of 2D Objects

4m

11m

8 cm

10cm

10 km

1.3 Surface Area of Objects Made from Right Rectangular Prisms

10 cm

10 cm

10 cm

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Strategy #1

Strategy #2

Each of you have been given a different model made from blocks.

Using a piece of graph paper, make a 2D drawing of the front, back, top, bottom, left side, and right side of your model.

Tips to make it easier:- Make you drawing so that one square on the graph paper equals one block- You may write "front", "back" etc on your model as long as

you erase it after

When finished, assume one face of each cube has an area of 1 unit2 and calculate the surface area of the model you were given.

September 02, 2014


There are several different strategies to determine the surface area of composite objects made from rectangular prisms.

1. Count the number of exposed faces from a model or;

2. Create a top, bottom, front, back and side views and determine the area of each then add together or;

3. Determine the total surface area of each block and subtract the overlapping areas.

Method 1:

Method 2:

Method 3:

September 02, 2014

50 m

114 m

50 m

50 m

50 m

50 m

50 m

30 cm

15 cm

17 cm

6 cm 6 cm

6 cm

September 02, 2014

30 cm

15 cm

17 cm

6 cm 6 cm

6 cm

50 m

114 m

50 m

50 m

50 m

50 m

50 m

September 02, 2014

50 m

114 m

50 m

50 m

50 m

50 m

50 m

30 m30 m

30 m

How would it change if this part was added?

Would sketching side top, bottom, front, back, and side views work to calculate surface area for this object?

50 m

114 m

50 m

50 m

50 m

50 m

50 m

30 m30 m

30 m

What if it was a building and had 4 m by 15 m door... and only the outside walls needed to be painted?

September 02, 2014

1.4 Surface Areas of Other Composite Objects

Review of Area formulas:

Area of a Rectangle:

Area of a Triangle:

Area of a Circle:

Also, circumference will be important: c = πd or c = 2πr

A = bh

A = bh 2

A = πr2

Calculate the total surface area of this cabin

5 m8 m

3.5 m

4 m

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10 cm

14 cm

15 cm

20 cm

50 cm

September 02, 2014

One last one... and it's a tough one!

50 cm14 cm

40 cm

14 cm

40 cm

16 cm

20 cm16 cm

Excluding the bow how many cm2 of icing is required for this cake? (There is icing on the top of each layer, but not the bottom)

Calculate the total surface area of this cylinder:

18 cm

80 cm

chapter 1: prior skills & knowledge

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