chapter 11 – part a lesson’s covered: 11.1 11.2 quiz covering 11.1-11.2 11.4 11.5 quiz covering...

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