chapter 11 – part a lesson’s covered: 11.1 11.2 quiz covering 11.1-11.2 11.4 11.5 quiz covering...
TRANSCRIPT
CHAPTER 11 – PART ALesson’s Covered:11.111.2•QUIZ COVERING 11.1-11.211.411.5•QUIZ COVERING 11.4-11.5Part A TEST
Algebra I – Chapter 11Daily Warm-Up
Factor the Trinomial1. y² + 5y - 14
Solve the Quadratic Equation by Factoring2. x² - 9x = -14
11.1 Ratio and Proportion
Objectives:1. Write and solve proportions
2. Use proportions in real life
The steepness of a hill can be written as the ratio of its height to its horizontal extent.
Ratios can be expressed as follows:
2 to 3
2:3
2/3
A ratio is a comparison of two quantities of the same kind, expressed in the same units.
A proportion is an equation stating that 2 ratios are equal.
Read: “1 is to 3 as 4 is to 12”
Read:
dc
ba
y6
x5
36
5a
is a proportion124
31
Cross Product Property: The product of the extremes equals the product of the means.
64
32
Apply the cross product property to:
dc
ba
y6
x5
36
5a
1212
bcad 6x5y 303a
4362
Extremes Means
a = ?
Solve the proportions:
w8
32
y3
64
52t
t10
12 w2w = 24
4y = 18
50 = 2t²
25 = t²
29 y
5 t
Solve the proportions:
5-3-x
43x
23-d
d5
YOUR TURN-Solve the proportions:
x9
4x
2 x
3
1 x
4
Extraneous solution – solution that doesn’t satisfy the original equation.
23y
3y9y2
Why not ±3?
2(y² - 9) = (y + 3)(y – 3)
2y² - 18 = y² - 9
y² - 9 = 0
(y + 3)(y – 3) = 0
y = ?
Remember to check in original equation!!
y
8
4
y y
Extraneous solution – solution that doesn’t satisfy the original equation.
YOU TRY-Remember to check for Extraneous Solutions!
Using Proportions in Real Life
You want to make a scale model of a parade float.The float is 5.5 feet long and 10 feet high. Yourmodel will be 14 inches high. How long should it be?
Algebra I – L 11.1
What is present in every paint bucket but absent in every translucent bucket?
Clue: it is not paint.
Tonight’s Homework:Pages 646-647#’s 17-18, 27-28, 30-31, 44
Algebra I – Chapter 11
Daily Warm-Up
2
3
x x
x x
2 x
3
1 x
4
x 5
10 2x
11.2 Percents
Goal:Use equations to solve percent problemsUse percents in real-life problems
What is 50% of 90?
Percent Equation: Percent Proportion:
a = p * b a/b = p/100
a = b = p =
of = . (multiplication)
% = decimal (move 2 to the left) – For Equation ONLY!
Is = “=“
Let’s Solve for a, p, or b?
What is 150% of $200?
9.6 is 12% of what number?
131 is what % of 255?
Your Turn. Solve each % Problem.
18 is what percent of 60? 52 is 12.5% of what number?
What distance is 24% of 710 miles?
$4 is 2.5% of what number? 2 is what percent of 40 feet?
9 people is what percent of 60 people? 85% of 300 is what number?
25.9%
14.9%
9.5%7.9%
10.2%
31.6% School
Eating/Dressing
TV
Homework
Other
Sleeping
The graph shows the results of a poll of student taken to find out the average amount of time spent on various activities in a 24 hour period.
How many hours on average spent sleeping?
How many hours watching TV?
Algebra I – L 11.21-2-3-4-5-6
I am a 6 letter word.Letters 6-5-2 spell out a drink.Letters 4-5-2-3 spell out a fruit.Letters 1-2-6 spell out a pet.Letters 3-2-6 spell out a pest, which often gets eaten by 1-2-6.
What am I?
Tonight’s Homework:Page 653#’s 10-20, 33
Algebra I – Chapter 11
Daily Warm-Up What number is 54% of $88?
631 feet is what percent of 1,281 feet?
11.4 Simplifying Rational Expressions
Goals:1.Simplify a rational expression2.Use rational expressions to find geometric probability
Rational number = a number that can be written as a quotient of 2 integers (fraction)
43
3612
3045
125625
19664
Rational expressions = an expression number that can be written as a fraction of 2 nonzero polynomials
5)2(x2x
2x6)x(x
x4)x(x
5xx
4x
x
x 6
Simplify Rational Expressions:
1. Factor out a GCF, if possible
2. Factor if it is quadratic, if possible
3. Reduce the numerical part, if possible
4. Cancel out common factors (blocks)
4)(xx4)x(x
2
5xx)-5(3
5)4x(x5)(x2x2
3)5(x3)(5x
2
2
6x6x2x
10x5x15x2
=
=
What values for x would make these expressions undefined?
Simplify Rational Expressions:
1. Factor out a GCF, if possible
2. Factor if it is quadratic, if possible
3. Reduce the numerical part, if possible
4. Cancel out common factors (blocks)
6x-12x4x8x
2
23 )12(6
)12(4 2
xx
xx=
=9x
96xx2
2
x ≠ 0, 1/2
What values for x would make these expressions undefined?
Simplify Rational Expressions:
1. Factor out a GCF, if possible
2. Factor if it is quadratic, if possible
3. Reduce the numerical part, if possible
4. Cancel out common factors (blocks)
xx34xx
2
2
Simplify Rational Expressions:
1. Factor out a GCF, if possible
2. Factor if it is quadratic, if possible
3. Reduce the numerical part, if possible
4. Cancel out common factors (blocks)
2110xx12-4x
2
x
x
49
28 262
4
x
x
Simplify Rational Expressions:
1. Factor out a GCF, if possible
2. Factor if it is quadratic, if possible
3. Reduce the numerical part, if possible
4. Cancel out common factors (blocks)
2
23
6x9x3x
10x5x 45 -15x
2
Simplify Rational Expressions:
1. Factor out a GCF, if possible
2. Factor if it is quadratic, if possible
3. Reduce the numerical part, if possible
4. Cancel out common factors (blocks)
Algebra I – L 11.4There were four brothers who were born in this world together.
One runs but is never weary, One eats but is never full, One drinks but is never thirsty, One sings a song that is never good.
Who are they?
Tonight’s Homework:Page 667#’s 9-11, 18-20, 25-26
Algebra I – Chapter 11
2
6 2
x 5x + 6
x
Daily Warm-Up
11.5 Multiplying and Dividing Rational Expressions
Goals:
Multiply and divide rational expressionsUse rational expressions in real-life models
Multiply fractions:
dc
ba
Divide fractions:
dc
ba
bdac
cd
ba
bcad
92
46
92
46
Keep
Change
Flip
Multiply/Divide Rational Expressions:
1. If Division, change to Multiplication
2. GCF & factor quadratics if possible
3. Cancel all common factors (blocks)
14n10n
5n7n 3
2
5
Multiply/Divide Rational Expressions:
1. If Division, change to Multiplication
2. GCF & factor quadratics if possible
3. Cancel all common factors (blocks)(2x-3)(x+1)
3)(2x3x2x
12x2
1
Multiply/Divide Rational Expressions:
1. If Division, change to Multiplication
2. GCF & factor quadratics if possible
3. Cancel all common factors (blocks)
2
y y - 5
y 25 y 5
Multiply/Divide Rational Expressions:
1. If Division, change to Multiplication
2. GCF & factor quadratics if possible
3. Cancel all common factors (blocks)
32 2x2x
4 4x x8x
Multiply/Divide Rational Expressions:
1. If Division, change to Multiplication
2. GCF & factor quadratics if possible
3. Cancel all common factors (blocks)
5n2-n
2n2-n
=
2n
5+n2n
2-n
• =n+5
2n
keep
changeflip
Multiply/Divide Rational Expressions:
1. If Division, change to Multiplication
2. GCF & factor quadratics if possible
3. Cancel all common factors (blocks)
4)-(x5x20x-5x²
=
5x9xx
5x4x 2
Now you try:
1)(x33x
2xx 22
Algebra I – L 11.5
Tonight’s Homework:Page 673#’s 12, 15, 18, 22, 32
1. Remove six letters from this sequence to reveal a familiar English word.
BSAINXLEATNTEARS
1. If you drop me I'm sure to crack but give me a smile and I'll always smile back
CHAPTER 11 – PART BLesson’s Covered:11.611.7•QUIZ COVERING 11.6-11.711.8Part B TEST
11-6 Adding and Subtracting Rational Expressions
Goal: Add & subtract rational expressions with like and unlike denominators.
Vocabulary
LCD – Least common denominator is the least common multiple of the denominators of two or more fractions.
Adding and Subtracting with Like Denominators
Let a, b, and c be polynomials, with c ≠0. To add, add numerators:
a + b = a + bc c c
To subtract, subtract the numerators.a – b = a – bc c c
Ex 1 – Common Denominators
7 + 2x -7 = 7 +(2x – 7) = 2x = 2x 2x 2x 2x
5 - 2m = 5 – 2m 3m – 4 3m – 4 3m - 4
1
Now you try:
1) 5 + x – 6 =3x 3x
2) 9 - 4n = 2n – 1 2n - 1
Ex 2 – Common Denominators
3x - x + 1 = 3x – ( x + 1)
2x² + 3x - 2 2x² + 3x - 2 2x² + 3x – 2
= 2x – 1 Factor and divide out common factors
(2x – 1) ( x + 2)
= 1 x + 2 Simplified form
Add or Subtract and Simplify
57x2x33x
57x2xx1
22
2x3x2x
2x3x4x
22
Finding the Least Common Denominator (LCD)
Step 1. Factor each denominator completely.
Step 2. The LCD is the product of all unique factors each raised to the greatest power that appears in any factored denominator.
Adding or Subtracting Rational Expressions with Different Denominators
Step 1. Find the LCD.Step 2. Rewrite each rational expression as
an equivalent rational expression whose denominator is the Step 1 LCD.
Step 3. Add or Subtract numerators. Write result over LCD.
Step 4. Simplify the resulting rational expression.
Example 3
1. Find least common denominator2. Rewrite fractions as LCD
fractions3. Add or Subtract4. Reduce
3 + 1 4x 6x²
Example 4 - TOGETHER
x + 3 - 8 = x - 2 x + 2
YOU TRY…Add/Subtract and Simplify
1) 3 + 4 x + 3 x – 3
2) 2 - 1x – 1 x + 4
Algebra I – L 11.6
What starts with T ends with T and is full of T?
Tonight’s Homework:Page 679#’s 10-11, 18-19, 28
Algebra I – Chapter 11
Daily Warm-Up
3 + 4 x + 3 x – 3
Lesson 11.7Dividing Polynomials
1. Divide a polynomial by a monomial or by a binomial factor.
2. Use polynomial long division.
Dividing a Polynomial by a Monomial
LONG DIVISION REVIEW
Divide 5103 by 7 using long division.
Dividing a Polynomial by a Monomial
Divide 12x² + 15x – 18 by 3x
Dividing a Polynomial by a Monomial
Divide 12x² – 20x + 8 by 4x.
Divide 9x³ – 27x² + 21x – 18 by 3x².
Long Division
Divide 270 by 20
)270201(20)=20
Subtract and bring down next digit
2070
3(20)=6060Subtract. The remainder is 1010
1Dividend
Divisor
3 10/20
The quotient is 13 ½.
remainder
divisor
Polynomial Long Division
Put dividend in standard form (with spacers) and into the division box
Multiply the divisor so the first terms are exact
Draw the line, change the sign, and combine
Bring down next term With no more terms, put the
remainder in a fraction with the divisor.
Polynomial Long Division
Divide x² - 3x + 5 by x + 2.
YOU TRY - Polynomial Division
Divide x² + 2x + 4 by x – 1.
YOU TRY - Polynomial Division
Divide 5n² + 2 by n + 1.
YOU TRY - Polynomial Division
Divide x – 12 + 3x² by 3x + 1.
Algebra I – L 11.7
I can easily be broken, yet, no one touches me. What am I?
Tonight’s Homework:Pages 687-688#’s 18-20, 27-29, 48
Algebra I – Chapter 11
Daily Warm-UpDivide 6x³ – 24x² + 20x – 10 by 2x
Divide x² - 3x + 2 by x – 2
11.8 Rational Equations
Goal:Solve rational equations.
A rational equation is an equation that contains rational expressions.
Cross Multiplying - Can be used only for equations with a single fraction on each side.
3y
2y5
4-x4
3x
Now you try:
Multiply by the LCD to get rid of fractions. Then solve.
x4
31
x2
x1
153
5x2
You may need to factor first to find the LCD. Then solve as before.
12yy10-
13y
4-2
(y+4)(y-3)
LCD=(y+4)(y-3)
YOU TRY…
149xx8
17-x
32
YOU TRY…
1 2 1
2 xx
YOU TRY…CHALLENGE!
2 7 1
3 3x - 12 x - 4
Summary
3y
2y5
x4
31
x2
12yy10-
13y
4-2
Cross Multiply
Multiply by the LCD
Factor and then multiply by the LCD
Algebra I – L 11.8
What is the animal who's name is three letters long, take away the first letter and you have bigger animal?
Tonight’s Homework:Page 694#’s 14-15, 21, 28, 33