Unit 1: Square Roots and Surface Area
1.1. Square Roots of Perfect Squares
Perfect Squares
Recall the perfect square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, etc. These are the numbers that can be written as a product of two equal factors. They can be the area of a square.
Fraction Perfect Squares
One way to determine whether a fraction is a perfect square is if we can create a square with that area.
Example:
Also note the diagram of on the bottom right p. 6
A fraction in simplest form is a perfect square if it can be written as a product of two equal factors.
Example: is a perfect square since
Also, if the numerator and denominator are perfect squares when the fraction is in lowest terms, it is a perfect square.
Example: and are perfect squares
Decimal Perfect Squares
Example: is a square number
Note that is not a perfect square since
Determining the Square Root of Fractions
Recall that squaring is connected to the area of a square while square roots are connected to the side length of a square.
To determine the square root of a number, you can use the area model: draw a square with that area. The square root is the side length of the square.
See the diagram of above. You can see that since is the
side length of the square with an area of .
Also note that since is the side length of a square with
area of . See the diagram bottom right p. 6.
You can also use the definition of a perfect square. If you can write a fraction as the product of identical factors, that factor is the square root.
Example:
Finally, you can find the square root of a fraction by taking the square root of both the numerator and denominator. This is perhaps the best method for large numbers though it may still require the use of a calculator.
Determining the Square Root of Decimals
We can use two of the methods outlined above – the area model or the definition.
Example: since we can create a square with an area of and the side length is .
Example:
Finally we could change a decimal number to fraction form and use either method outlined above.
Patterns
We all know that . Notice the pattern below:
Here we see a pattern emerging. Notice that there are exactly half as many places before/after the decimal after we take the square root.
Example: To find the value of :
Determine the number of places after the decimal, 3 in this case. So
Note 1:
The square root of numbers greater than 1 is less than the number.
The square root of numbers between 0 and 1 will be greater than the number.
Note 2:
The square root of any perfect square number will either terminate or repeat.
Ex. terminating
repeating
We can use this idea to determine whether a number is a perfect square as well. If we use our calculator to determine the square root of a number and the answer is not repeating or terminating (meaning it goes on forever without the same digits repeating) then the number is not a perfect square.
Note 3:
We should note that both and both equal 9. That means that there are always two roots, a positive and a negative.
However, Mathematicians use the radical symbol ( ) to denote the positive or principal square root since many real life applications only require the positive answer. When the question is the answer is 3.
Common Error
Go through pp. 9-10 Examples 1-3.
1.2. Square Roots of Non-Perfect Squares
Not all fractions and decimals are perfect squares. That is, they can not be written as the product of identical factors. Such numbers are called non-perfect squares. Therefore it is necessary to estimate the square root of such numbers.
Method 1: Benchmarks
Ex 1. Here we chose a fraction whose numerator and denominator are close to the numerator and denominator that we are starting with and then found the square root.
Ex 2. Here we chose a fraction that was close but easier to work with before we found the square root.
Let’s look at some common perfect square decimals and their square roots:
Ex 3.
a) Estimate to one decimal place.
Solution:
b) to 2 decimal places
is between and .
Using a number line we can see that because it is more than 0.65 but not really close.
Here we determine what two perfect squares the number is between and then used a number line to determine where the answer lies in relation to the others.
Ex 4. is between and .
Here we chose two perfect squares that surrounded the number we were trying to take the square root of and then using a number line we can see that because it is just less than 1.15, the halfway mark.
Method 2: Calculator.
The more decimal places we use the more accurate our approximation will be. However, we generally limit our answers to one or two decimal place.
Also see example 1 pp. 15-16
0.36 0.45 0.49
0.6 0.65 0.7
Note:
There are an infinite number of values that have square roots between any two rational numbers
Ex: What numbers have square roots between 2.3 and 2.31?
Any numbers between and will do.
, , , and are just a few examples
See example 2 p. 17
The Pythagorean Theorem
Complete p. 19 #13b & d together
Also see example 3 pp. 17-18
Complete pp. 18-20 #4ace, 5ace, 6*, 7*, 8, 9**,10, 11*,12*, 13ac, 16*, 17, 19, 20a
*Do not rely on calculators
**Recall that roots of numbers between 0 and 1 will be larger that number you are taking root of.
To find the length of the hypotenuse use: (hyp)2 = (leg 1)2 + (leg 2)2
1.3. Surface Areas of Objects Made from Right Rectangular Prisms Complete Investigation p. 25 & Connect p. 26 together
A Composite Object is made up of, or composed of, other objects.
Surface Area of Composite Objects Made from Cubes Method 1
Count all visible squares on each of the six faces.
Multiply by the area of each square.
See example 1 p. 27 (left solution)
Method 2
Determine the total surface area of all cubes by multiplying the number of cubes by 6.
Subtract the faces that are overlapped. Remember that for every overlap two faces are hidden.
Multiply by the area of each face.
See example 1 p. 27 (right solution)
Complete pp. 30-31 #4,7*
Recall two formulas: For Rectangles
For Rectangular Prisms
Have students complete Skill builder from homework book
Surface Area of Composite Objects Made from Right Rectangular Prisms
Here we determine the surface area of each individual prism and then subtract the overlap. Again, we must remember that for each area of overlap, two surfaces are “hidden” and must be taken away.
See example 2 p. 26 of homework book on smartboard
Complete #1-5 from practice book
Go through examples 2 & 3 pp. 28 & 29 together on smartboard
Discuss #1 p. 30
Complete pp. 31-32 #8, 10*, 11*
1.4. Surface Areas of Other Composite Objects
Recall how to find the surface area of a Right triangular prism and a Cylinder
Complete Skill Builder pp. 33-35 of homework book
Go through examples 1& 2 from the homework book
Go through example 2 pp. 36-37 in the text
Note 1: With cylinders always use the π button on your calculator in place of 3.14
Note 2: We always keep context in mind! Find the total surface area unless restricted by the context.
Additional Example: Todd makes a two-layer cake. He puts strawberry jam between the layers instead of icing. He plans to cover the outside of the cake with icing. Calculate the area that needs icing.
Solution 1:
Rectangular Prism Bottom
25 x 25 = 625 Top (no icing on bottom)
4 x 5 x 25 = 500 Front, Back, Left & Right
1125 cm2 Total surface area
Cylindrical Top
2 x 3.14 x 102 = 628 Circles
2 x 3.14 x 10 x 5 = 314 Lateral Surface
942 Total Surface area
Omit
2 x 3.14 x 102 = 628 Circle overlap
Surface Area = 1125 + 942 – 628 = 1439 cm2
Solution 2:
Rectangular Prism Bottom
25 x 25 = 625 Top only
4 x 5 x 25 = 500 Front, Back, Left & Right
1125 cm2 Total surface area
Cylindrical Top
2 x 3.14 x 10 x 5 = 314 Lateral Surface only
What if we do not have a right triangle and we do not have the height? We will need to use the Pythagorean Theorem to calculate the height.
Example: The base of a triangular prism is given below. It is an equilateral triangle with sides 6 cm. Find the area of the triangle.
= 62 - 32 So
= 36 - 9
= 27 cm2
=
Complete pp. 40-43 #3ad, 4, 5, 6, 7a, 9
Also questions from homework book if necessary for some students
6 cm
