Download Unit 1 – Square Roots and Surface Area Section 1.1 – Square Roots

Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
Unit 1 – Square Roots and Surface Area
Section 1.1 – Square Roots and Perfect Squares
Review: Changing Decimals to Fractions
To change a decimal to a fraction, we move the decimal to the right, and put the
number over a power of 10.
For example, let’s look at 0.6.
To get rid of the decimal we would move it one place to the right. Because we move
the decimal one place to the right, we add one zero on the bottom.
Therefore,
Try the following:
a) 0.08
b) 0.25
c) 0.379
Recall, a perfect square is the result of a whole number multiplied by itself. For
example, 16 is a perfect square since
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
To determine if a decimal is a perfect square, we can change it into a fraction, or
remember this rule:
Rule: If the decimal has an even number of decimal places and consists of a perfect
square, the decimal number is a perfect square.
For example,
a) 0.49 has an even number of decimal places (two numbers after the decimal)
fcp://@fc.nlesd.ca,%2318870007/Mailbox/004b262b-011feef7and 49 is a perfect square,
therefore 0.49 is also a perfect square.
b) 0.00016 is NOT a perfect square. Even though 16 is a perfect square it has an
odd number of decimal places (three numbers after the decimal).
Some fractions can also be perfect squares. In order for a fraction to be a perfect
square, both the numerator and denominator have to be perfect squares.
Let’s consider the following situations.
a) Is
a perfect square?
We can also be show this by drawing a diagram using squares.
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Unit 1 – Square Roots and Surface Area
b) Is 16
c) Is
Grade 9 Mathematics
a perfect square?
a perfect square?
You can also check to see if a number is a perfect square in your calculator. If the
square root terminates (ends) it is a perfect square.
Decimal
Value of Square Root
Type of Decimal
Is it a perfect square?
1.69
3.5
70.5
5.76
0.25
2.5
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
Try the following:
.
1.
Use a diagram to determine the value of
2.
Which of the following are perfect squares? Explain.
3.
Find each square root:
a)
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b)
c)
d)
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Unit 1 – Square Roots and Surface Area
4.
Which of the following are perfect squares?
a) 1.96
5.
Grade 9 Mathematics
b) 0.9
c) 0.036
Calculate the number whose square root is:
a) 0.6
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b) 0.04
c) 1.4
d)
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
Section 1.2 – Square Roots and Perfect Squares
Recall:
To estimate a perfect square, we locate the perfect square just above and just below
the number you are looking for.
For example,
=5
a)
=4
b)
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
We can use the same way we estimate non-perfect square whole numbers, to
estimate the square root of non-perfect fractions and decimals.
Consider
.
Between which two decimal perfect squares does 0.38 fall?
0.6
0.7
Example 1:
Estimate each square root and show workings.
a)
b)
c)
b)
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
To estimate a fraction, we find the closest perfect squares to the numerator and the
denominator and find the square root.
Consider
.
The closest perfect square to 8 is 9. The closest perfect square to 15 is 16.
Therefore, we can estimate
as
which is then equal to
.
Try the following:
a)
b)
We can also estimate fractions by changing the fraction to a decimal first, and then
estimating.
Consider
.
is the same as 0.3.
Since 0.30 falls between the perfect squares 0.25 and 0.36,
and
so we estimate in between these numbers.
0.5
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falls between
0.6
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
Using a method of your choice, estimate each of the following:
Find each square root:
a)
b)
c)
Recall: Pythagorean Theorem
A right triangle is a triangle with one 90o angle. The Pythagorean Theorem states
that for any right triangle, the area of the square on the hypotenuse is equal to the
sum of the area of the squares on the other two sides (legs).
The side directly across from the 90o angle is
called the _______________________________.
This is always the longest side in a right triangle.
The two remaining sides are called
______________________________.
c2
a2
b2
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
Example 1:
a) Find the missing value:
a)
c
b)
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
Section 1.3 – Surface Area of Objects Made from Right
Rectangular Prisms
Investigation
Assume each face of a linking cube has 1 unit2.
What is the surface area of 1 cube? Now link 2 cubes together and determine the
surface area. Continue to do so, linking each additional cube to the end creating a
“train”.
Fill in the chart below.
Number of Cubes
Surface Area (square units)
1
2
3
4
5
What happens to the surface area each time you add a cube? Why do you think it
changes this way?
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
Here is an object made from 4 cubes. What is the surface area? How did you
determine the surface area?
We could easily find the surface area by finding the total surface area of each
block, adding them together and then subtracting the _______________________!
Find the surface area of the object below:
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
Surface Area of Square/Rectangular Prisms
To find the surface area of any prism, we need to individually find the area of all
surfaces and add them together. In the case of a rectangular prism, all the faces are
rectangles so we use the same formula for all six rectangles.
Formula for the Area of a Rectangle: _________________________________
Example 1:
8 cm
Front/Back:
10 cm
6 cm
Top/Bottom:
Side/Side:
Total Surface Area:
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
Try these:
a)
5 cm
6 cm
8 cm
b)
12 cm
12 cm
12 cm
A _____________________________________ object is composed of two or more objects.
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
To find the surface area of a composite object, imagine dipping the object in paint.
What happens when you separate the blocks below once they have been painted? –
You will have an area on each block that is not painted because it was touching the
other block. This unpainted part is called the overlap.
To find the surface area, we calculate the area of all the faces covered in paint. The
overlap is not painted, so it is not part of the surface area; thus we find the total
surface area of each block, add them together and subtract the overlap from the
total at the end.
Example 1:
1.
First calculate the surface area of the larger prism.
Front/Back:
10 m
6m
Top/Bottom
2m
4m
5m
2m
Side/Side:
Total Surface Area:
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Unit 1 – Square Roots and Surface Area
2.
Grade 9 Mathematics
10 m
Second, calculate the surface area of the smaller prism.
Front/Back:
6m
2m
4m
Top/Bottom
5m
2m
Side/Side:
Total Surface Area:
10 m
3.
6m
Now calculate the overlap.
2m
2m
4m
2m
5m
4m
Total Surface Area of the Composite Figure:
________________________ + ________________________ - ______________ = ________________
SArea of Large Object + SArea of Small Object – 2 Overlap
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
Try These:
1)
2)
6
4
1
4
2
4
4
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
Section 1.4 – Surface Area of Other Composite Objects
To find the surface area of a triangular prism, we need to use two formulas.
Area of a Rectangle =____________________
Area of a Triangle = ___________________
Example 1:
2 Triangular Faces:
3 Rectangular Faces:
 
First:
Second:
Third:
Total Surface Area:
  
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
Try These:
A)
B)
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
Finding the Area of a Composite Object
When finding the surface area of a composite figure involving triangular prisms, we
use the same process we did before.
Step 1: Calculate the surface area of the larger prism.
Front/Back :
Top/Bottom:
Side/Side:
Total Surface Area:
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
Step 2: Calculate the surface area of the smaller prism.
2 Triangular Faces:
3 Rectangular Faces:
 
First:
Second:
Third:
Total Surface Area:
Step 3: Calculate the overlap.
What is the shape of the overlap?
Be careful with these composite
objects the overlap could be a
triangle OR rectangle.
Area of Triangle:
Total Surface Area:
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
Try These:
A)
B)
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
Recall:
Surface Area Formula for a Cylinder
Example 1:
Two cakes are arranged as shown below. These cakes are to be covered in frosting.
What is the area of the frosting?
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Unit 1 – Square Roots and Surface Area
Grade 9 Mathematics
Example 2:
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