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Transcript of IAP20S
Factors & Products
Factors and Products
Roots and Powers
Relations and Functions
are measurement units such as the inch, foot, yard, and mile
commonly used in the construction industry
The Imperial System
Construct Understanding, p.5
Relationship between Imperial units
1 ft = 12 in
1 yd = 3 ft = 36 in
1 mi = 5280 ft = 1760 yd
used to estimate a measure
A fraction of an imperial measure of length is written in fraction form, not decimal form.
converting a measurement from one unit to another
Convert 7 yd to:
Convert 62 in to:
feet and inches
yards, feet, and inches
There are 2 types of measurement systems
1. Imperial system
2. SI system (metric system)
Why is it necessary to have standard units of length?
to ensure everybody gets the same measure for the same length
count the number of divisions
1 / 16 inch
Ben buys a baseboard for a bedroom. The perimeter of the bedroom, excluding closets and doorway is 37 ft.
a. What length of baseboard is needed, in yards and feet?
b. The baseboard material is sold by the yard. It costs $5.99/yd. What is the cost of material before taxes?
is used to verify a conversion between units
convert between imperial units by multiplication or division
set up and solve a proportion that equates two ratios
Winter plans to run for school president. She plans to make banners that are 33 in. long. Her friend bought a 15 yd. fabric.
a. how many banners can they make?
b. use unit analysis to check the conversions.
International system of units
modern form of the
works by using
Convert 15 m to:
Construct understanding, p. 14
Conversions between SI and Imperial units
1 inch = 2.54 cm
1 foot = 30.48 cm
1 yard = 0.914 m
1 mile = 1.61 km
1 mm = 0.04 in
1 cm = 0.39 in
1 m = 39.37 in
= 3.28 ft
= 1.09 yd
1 km = 0.621 mi
A flag pole measures 22 m long. What is the measurement to the nearest foot?
After meeting in Winnipeg, Autumn drow 53 miles north, and Dan drove 89 km south. Who drove farther?
Ace knows that he is 5 ft 8 in tall.
a. Convert his height to centimetres.
b. Use mental math and estimation to justify that the answer is reasonable.
A truck driver knows that his load is 15 ft. wide. Regulations along his route state that any load over 4.3 m wide must have wide-load markers and an escort with flashing lights. Does the truck need markers?
The Pythagorean Theorem
in a right triangle...
a + b = c
Surface area: Right Pyramid
A = b + 2bs
Surface area: Right cone
A = 1 s (perimeter of base) + base area
* for a right square pyramid
= area of the triangular faces
shortest distance on the curved surface between the apex and a point on the circumference of base
SA = rs + r
is a triangular pyramid
amount of space an object occupies
amount of material a container holds
A right cone has a base radius of 3ft and a height of 6 ft. Calculate the SA to the nearest square foot.
Volume: Right Prism
Volume: Right Pyramid
2 congruent and parallel bases
lateral faces are rectangular
V = base area x height
The V of a right pyramid is 1/3 the V of a right prism (w/ same base and height)
V = 1 Ah
Right Rectangular Prism
Right Rectangular Pyramid
V = lwh
V = 1 lwh
Volume: Right Cylinder
Volume: Right Cone
V = r h
Surface area: Right Prism
SA = sum of all areas of all faces
For a right rectangular prism,
SA = 2 (lw + wh + lh)
Surface area: Right Cylinder
SA = 2 rh + 2 r
has two parallel, congruent, circular bases
The V of a right cylinder is 3 times the volume of a right cone (with same base and height)
V = 1 r h
Surface Area: Sphere
every point on its surface has the same distance from the centre point
SA = 4 r
V = 4 r
SA = SA of one half a sphere + area of circle
V = volume of one-half of sphere
1.7 Solving Problems involving Objects
composed of 2 or more distinct objects
Volume of a composite object
identify the distinct objects
add the volumes of the objects
Surface area of a composite object
identify the faces of the composite object
calculate the sum of the areas of these faces
comes from Greek words
"tri +gonia + metron"
three angle measure
The sides of a right triangle is named in relation to the reference angle:
opposite: the side opposite to the given angle (b)
adjacent: the side beside the given angle (a)
hypotenuse: the side opposite the right angle (c)
In a given acute angle,
NOTE: All interior angles add up to 180
The primary trigonometric ratios
Determining measures of
You can use the
inverse trig ratios
to find a measure of an angle.
Angle of inclination
Angle of depression
These angles are formed between a horizontal line and the line of sight.
Solving the triangle
to determine the measures of all angles and the lengths of all sides in a right triangle
Solving problems using trig ratios
Solve this triangle.
Solve this triangle.
What do we know about triangles?
The two acute angles add up to 90
1. Draw and label the triangle
2. Determine the ratio to use
or you might need to use the Pythagorean theorem first
(you might need to make assumptions)
Solving problems involving more than one right triangle
You see this man on top of a tree as you were walking in the park. When you are 100 m from the tree, the angle of inclination is 30 . About how high is the man? Any assumptions?
1. Draw and label a diagram/triangle
Find CD and BD.
tan = opposite
(the ratio between the opposite side and the adjacent side)
If tan = 1.2, then the opposite side is 1.2 times longer than the adjacent side.
sin = opposite
(the ratio between the opposite and the hypotenuse)
If sin = 0.7, then the opposite side is 0.7 times longer than the hypotenuse.
cos = adjacent
(the ratio between the adjacent side and the hypotenuse)
if cos = 0.6, then the adjacent side is 0.6 longer than the hypotenuse.
A surveyor stands at a window on the 9th floor of a tower, 39 m from the ground. The angle of inclination to the top of a taller building is 31 and the angle of depression to the base of that building is 42 . Determine the height of the taller building.
a natural number having no factors expect 1 and itself.
the #s you multiply together to get another #
2 x 3 = 6
a counting number
first 10 prime numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
natural #s greater than 1 that are not prime
a factor that is a prime number
of a natural number is the number written as a product of its prime numbers
ex. Write the prime factorization of 3300.
ex. 2 Write the prime factorization of 2646.
draw a factor tree (p.135)
use repeated division by prime factors
Greatest common factor
of two or more numbers is the greatest factor the numbers have in common.
List all factors of:
ex 1. Find the GCF of 26 and 44.
find all factors of both numbers
then list all common factors
biggest -> GCF
ex 2. Find the GCF of 10 and 35.
Least Common Multiple
the least whole number that is a multiple of each of two or more numbers
ex 4. Determine the LCM of 9 and 6
list all multiples of each number until the same multiple appears
ex 5. Determine the LCM of 18, 30 and 40
write the prime factorization of each number
highlight the greatest power of each prime factor in any list
multiply the greatest powers of each prime factor
Solving problems that involve GCF and LCM
ex. What is the side length of the smallest square that could be tiled with rectangles that measure 8 in by 36 in?
: shortest, minimum, least
ex. What is the side length of the largest square that could be used to tile a rectangle that measure 8 in by 36 in?
other hints: least, biggest, maximum
list the prime factors of both numbers
then multiply the common prime factors
ex 6. Find the LCM of 18, 20 and 30.
Try this! Find the LCM of 27, 90 and 84.
any whole number that can be represented by an area of a square with a whole number side length
of a number (x), denoted , is a positive number whose square is x.
ex. Determine the square root of 7056.
ex. Determine the square root of 6084.
Write the prime factorization of the number
Rearrange the factors in 2 equal groups
any whole number that can be represented as the volume of a cube with a whole number edge length
of a number (n), denoted n, is a number whose cube is n.
area = 25 square units
side length = 5 units
root of a number
ex. Determine the cube root of 1331.
ex. Determine the cube root of 74088.
ex. Is 4500 a perfect square?
ex. Determine the cube root of 46656.
write its prime factorization
rearrange the factors in 3 equal groups
Common factors of a polynomial
any number that can be expressed as the ratio of 2 integers
have decimal representations that terminate or repeat
any number that cannot be expressed as the ratio of 2 integers
decimal representations neither terminate nor repeat
set up and solve a proportion that equates two ratios
A support cable is anchored to the ground 5m from the base of a telephone pole. The cable is 19 m long. It is attached near the top of the pole. What angle to the nearest degree does the cable make with the ground?
At a horizontal distance of 200 m from the base of an observation tower, the angle of elevation to the top of the tower is 8 . How high is the tower?
An observer is sitting on a dock watching a float plane in Vancouver harbour. At a certain time, the plane is 300 m above the water and 430 m from the observer. Determine the angle of elevation of the plane measured from the observer to the nearest degree.
From a radar station, the angle of elevation of an approaching airplane is 32.5 . The horizontal distance between the plane and the radar station is 35.6 km. How far is the plane from the radar station?
Systems of Linear Equations
A school district has buses that carry 12 passengers and buses that carry 24 passengers. The total passenger capacity is 780. There are 20 more small buses than large buses.
12s + 24l = 780
s = l + 20
- system of linear equations in two variables (s and l)
- also called a linear system
A solution of a linear system is a pair of values of s and l that satisfy both equations.
Somebody said that the district has 35 small buses and 15 large buses. Is s/he right?
You can verify by substituting the known values of s and l into both equations.
GOAL: L.S. = R.S.
The perimeter of a Manitoba flag is 16 ft. Its length is 2 ft. longer than its width. Create a linear system to model this situation.
In Calgary, a school raised $195 by collecting 3000 items for recycling. The school received 5 cents for each pop can and 20 cents for each large plastic bottle.
5.2 Solving a Linear System Graphically
Remember: the solution of a linear system is the coordinate pair of values that satisfy both equations.
If you were to graph both equations, the
coordinates (x, y) of the point of intersection
are the solution of the linear system.
Solve this linear system
2x + 3y = 3
x - y = 4
Wayne received and sent 60 messages on his cell phone in one weekend. He sent 10 more messages than he received.
How many text messages did Wayne send and how many did he receive?
p.409 #s 3, 5, 13
5.4 Solving a Linear System by Substitution
We can use algebra to find an exact solution
Solve this linear system
5x - 3y = 18
4x - 6y = 18
Solve this linear system.
Create a linear system to model this situation and solve using elimination.
Nuri invested $2000, part at an annual interest rate of 8% and the rest at an annual interest rate of 10%. After one year, the total interest was $190.
p. 425 #s 4, 11, 12 ,16
7.5 Solving a Linear System
Adding or subtracting the two equations produces equivalent systems.
Solving by elimination
- solve a linear system by first eliminating one variable by adding or subtracting the two equations
Solve this linear system.
2x + y = -7
x + y = -4
Solve this linear system using elimination.
2x + 7y = 24
3x - 2y = -4
p. 437 #s 3, 8, 9,16
Write the prime factorization of 3990.
ex. A cube has a volume 4913 cubic inches. What is the surface area of the cube?
Try this! Find the GCF of 126 and 144.
p.401 #s 4, 6, 7, 13
Multiplying or dividing the equations in a linear system by a non-zero number does not change the graphs.
The result is called an equivalent linear system.
Each time Trisha went to the cafeteria, she bought either a bowl of soup for $1.75 or a main course for $4.75. During the school year, she spent $490 and bought 160 food items. How many times did Trisha buy soup? a main course?
7.6 Properties of Linear Systems
All linear systems we've studied so far only had exactly one solution.
Intersecting lines only have one solution.
Parallel lines don't touch (same slope, different y-intercepts). So there is no solution.
When lines have equal slopes and same y-intercept, they are coincident lines.
Coincident lines have an infinite solution.
Ex. 1 Determine the number of solutions of this linear system.
x + y = -2
-2x - 2y = 4
4x + 6y = -10
-2x - y = -1
Ex. 2 Given the equation -6x + y = 3, write another linear equation that will form a linear system with:
a. exactly one solution
b. no solution
c. infinite solutions
p. 448 #s 4, 7, 10, 11, 18