A skydiver jumps from 12,000 feet. Solve the equation h = 12,000 + (0.5)(-32.1)(576) to find his height after he free falls for 24 seconds.
Example
Same skydiver. Solve the equation h = 12,000 + (0.5)(-32.1)(144) to find his height after 12 seconds.
Multiplying Fractions
Example
Find each product.2 37 5
387
Example
Find each product.
1 133 7
3 54 7
Example
Find each product.
4 35
1 312 8
Multiplicative Property of -1
Example
Simplify
5 ( 2.2 )b y
2 ( 4.5 )x y
Example
Simplify
5 27 3
g h
2 35 5
m n
Closure Property
Assignments
#1 – due today P144: 2, 4 – 16
#2 – due next time P144: 18 – 44 even, 49 – 51, 53 – 58 Do multiplication in fraction packet
**You MUST write out each problem and show work on each! Calculators on for checking!!
4.2
Counting Outcomes
Tree Diagram• A diagram used to show the total number of
possible outcomes of an event
Counting Outcomes
• Outcomes – One possible result of a probability event
Ex – there are 16 outcomes for rock characteristics
• Sample Space – The list of all possible outcomes
Example
• Brooke is shopping for a new computer system. She has a list of 2 different CPUs, 3 different monitors, and 3 different printers. How many different ways can she choose one CPU, one monitor, and one printer from her list.
Example
• The Ice Cream Parlor offers the choices from the menu. Draw a tree diagram to find the number of different sundaes that can be made.
Events
• Event – A subset of possible outcomes
Ex – selecting a flavor of ice cream
Ex – selecting a topping
Counting
How many outcomes for sundaes was there?
How many choices for ice cream?
How many choices for toppings?
How many choices for whipped cream?
Is there any relation between those numbers?
Fundamental Counting Principle
Example
• How many different kinds of photo processing are possible?
Example
• How many different kinds of school sweatshirts are possible?
Assignments
• #1 – due today P148: 4 - 14
• #2 – due next time P148: 15 – 18, 20 – 23, 25 – 27, 29
P698 (4-1): 2 – 28 even
4.3
Dividing Rational Numbers
Dividing Rational Numbers
Example
• Find each quotient.• 8 ÷ (-2.5)
• -9.3 ÷ (-0.3)
• 16 ÷ (-2.5)
• -3.9 ÷ 3
• -8.4 ÷ (-1.2)
Multiplicative Inverse
• Multiplicative Inverse or Reciprocal –• Two numbers whose product is 1
3 4 14 3
15 15
Multiplicative Inverse Property
Dividing Fraction
Example
• Find each quotient.
2125
3 127 2
5 216 5
Example
• How much of each ingredient is needed to make one dozen cookies?
Example
• Evaluate if . 35
x
6x
2x
Example
• Evaluate if . 35
x
47x
Assignments
• #1 – due today• P156: 4 - 17
• #2 – due next time• P157: 18 – 34 even, 36 – 41, 51, 54, 56 – 63
• Finish division in fraction packet
Daily Agenda – Jan. 14
• 5-Minute Check
• Grade Assignment
• Quiz A
• 4.4 notes / assignment
4.4
Solving Multiplication and Division Equations
Division Property of Equality
Example
• Solve each equation. Check your solution.
5 30b
Example
• Solve each equation. Check your solution.
24 3g
Example
• Solve each equation. Check your solution.
5.5 22z
Example
• Solve each equation. Check your solution.
4 28t
Example
• Solve each equation. Check your solution.
0.1 7m
Example
• Brian received a $25 gift certificate from his grandparents for his birthday. How many $2.35 packages of trading cards can he buy with the gift certificate?
Multiplication Property of Equality
Example
• Solve each equation. Check your solution.
67w
Example
• Solve each equation. Check your solution.
192
m
Example
• Solve each equation. Check your solution.
2 85
x
Example
• Solve each equation. Check your solution.
84t
Example
• The manager of a movie theater estimates that 5/7
of the people who attend a matinee are children. How many people attended the 1:00 PM matinee today if 250 children’s tickets were sold?
Assignments
• #1 – due today P163: 1, 2, 4 - 10
• #2 – due next time P163: 12 – 32 even, 33 – 36, 38, 41 – 47, 49
4.5
Solving Multi-Step Equations
Procedure
• Work backwards Undo each operation
• Goal: Get the variable by itself
Example
• Solve each equation. Check your solution.
4 26x
Example
• Solve each equation. Check your solution.
3 12 27m
Example
• Solve each equation. Check your solution.
36.215
n
Example
• Solve each equation. Check your solution.
11 9 119v
Example
• Solve each equation. Check your solution.
5.2 38a
Example
• Solve each equation. Check your solution.
4 67
b
Consecutive Integers
• Integers in counting order 4, 5, 6, 7, 8…
• Consecutive even integers 2, 4, 6, 8…
• Consecutive odd integers 5, 7, 9, 11…
Example
• Find three consecutive integers whose sum is 27.
Example
• Find four consecutive odd integers whose sum is -8.
Example
• Find three consecutive even integers whose sum is -18.
Assignments
• #1 – due today P168: 4 - 12
• #2 – due next time P168: 4 – 32 even, 38, 44, 48
Daily Agenda – Jan. 26
• Turn worksheets into Inbox
• 4.6 notes / assignment
4.6
Variables on Both Sides
Goal
• Get the variables on the same side, then solve
Example
• Solve each equation. Check your solution.
8 9y y
Example
• Solve each equation. Check your solution.
9 4a a
Example
• Solve each equation. Check your solution.
2 165 5
x x
Example
• Solve each equation. Check your solution.
2 1 23 3
n n
Example
• Solve each equation. Check your solution.
Example
• In the 1996 Olympics, the winning times for the 100-meter freestyle were about 48.7 seconds for men and 54.5 seconds for women. Suppose the men’s times decrease 0.2 seconds per year and the women’s times decrease 0.3 seconds per year. Solve 48.7 – 0.2x = 54.5 – 0.3x to find when men and women would have the same winning times.
Solutions
• No solution – No value for the variable will make the equation
true Ex: 0 = 7
• Identity – Every value of the variable will make a true
equation Ex: x = x or 5 = 5
Example
• Solve each equation.
2 4 4t t t
Example
• Solve each equation.
16 7 16 16h h
Assignments
• #1 – due today P173: 4 – 15
• #2 – due next time P173: 16 – 32 even, 38 – 45
4-7
Grouping Symbols
Example
• Solve and check.
8 = 4(3x + 5)
Example
• Solve and check.
5(2x – 1) = -25
Example
• Solve and check.
5(h + 6) – 6 = 3(5h – 2)
Example
• Solve and check.
7 = 3(x + 1) - 2
Example
• Solve and check.
4(t + 5) + 6(2t – 3) = 12
Example
• The area of the trapezoid below is 64 square millimeters. Find the value of x.
Assignments
• #1 – due today P178: 4 – 10
• #2 – due next time P178: 12 – 30 even, 32, 35 – 38