DAY 1 CHAPTER FIVE...135 Opportunities To Learn DAY 1 REVIEW BLOCK 1. a) classify ABC b) with a protractor, find m ABC c) classify CBD d) find m CBD 2. List at least four facts that

134

DAY 1 CHAPTER FIVE What fascinating mathematics is now on our agenda? We will review the past four chapters a little bit each day because mathematics “builds.” Each concept is a foundation for another idea. We will have a “graph of the day” each day to reinforce what we know about graphing. The main focus of chapter five, however, is reviewing how to rewrite polynomials using the laws of exponents properly. In order to be successful, you have to know and understand the laws of exponents and how to use them properly. What is an exponent? An exponent is a number that tells how many times to use another number (or variable) as a factor. And factor implies “to multiply.”

5x means to use x as a factor five times, or

5x x x x x x

3( )a b means to use ( )a b as a factor three times, or

3( ) ( ) ( ) ( )a b a b a b a b

4(2 )c means to use 2c as a factor four times, or

4(2 ) (2 )(2 )(2 )(2 )c c c c c

But, 42c means to use c as a factor four times and multiply by ONE factor of 2,

That is, 42 2c c c c c

Recall that in the expression 42c : 2 is called the coefficient

c is called the base 4 is called the power or exponent Your teacher will provide you with some practice using exponents properly.

“If you can imagine it, you can achieve it. If you can dream it, you can become it.” - Anonymous

135

Opportunities To Learn DAY 1 REVIEW BLOCK

1. a) classify ABC b) with a protractor,

find m ABC

c) classify CBD

d) find m CBD 2. List at least four facts that you know about slope of a line. 3. What words are represented with the acronym ‘URESC’? 4. Graph Of The Day For the ‘Graphs of the Day’: put them on the same sheet of graph paper. Entitle the page “CHAPTER 5: GRAPHS OF THE DAY”. Label each graph with the day number. Call today’s “DAY 1.” Draw ONE pair of axes that are 8 units in each direction. Through the given points, draw and label the lines with the given slope.

Line a) through ( - 3, 5 ) with slope = 2

3

Line b) through ( 4, 0 ) with slope = 3 Line c) through ( -1 , - 5 ) with slope = 0

136

5. Make FLASH CARDS: EXPONENTS FRONT OF FLASH CARD BACK OF FLASH CARD

2

2 2 2 2 4

2 2 2 2 2

3 3 3 3 6

3 3 3 3 3

2 3 2 3 5

2 3 2 3 2 3

3 3 3 3 9

2 3 2 3 6

2

5 5 5

2

2

2

( ) ( )

( ) ( )

6 4 6 4 24

6 4 6 4 2

6 4 6 4 2

( )

a a a a a

a a a a a

a a a a a

a a a a a

a a a a a

a a a a a

a a a a a

a a a a a a

a a a

a a a

a a a a a

a a a a a

a a a a a

a a a

5 5

5 5 5 5 10

4 2 4 2 8

4 2 4 2 6

4 2 4 2 4 2

( ) 2

( ) ( ) ( ) ( )

( ) ( )

a a

a a a a a

a a a

a a a a a

a a a a a a

6. Practice the EXPONENTS flash cards a few times as we have been doing. Redo the missed ones as necessary and learn from them. 7. Correct the test on Chapter 4 to turn in tomorrow. DAY 2 Opportunities To Learn DAY 2 REVIEW BLOCK

1. 2 1

3 2

137

2. Use the ‘URESC’ method to solve the following. Show each step clearly. The measure of one angle is 30 degrees less than half the measure of the other angle. If the two angles are complementary, find the measure of each angle. 3. Solve for y in terms of x: 2x – 3y = 3 4. Graph Of The Day For the ‘Graphs of the Day’: put this on the same sheet of graph paper as you did DAY 1. Label this graph with the day number. Call today’s “DAY 2.” Draw ONE pair of axes that are 8 units in each direction. For each pair of points: i. Plot the two points ii. Find the slope of the line that contains these two points graphically. Show

your work. iii. Using the slope formula, algebraically find the slope of the line. Show your work on your graph paper. a) ( - 1, 5 ) ( 2, 3 ) b) ( - 4, - 5 ) ( - 2, - 1 ) 5. Practice the EXPONENTS flash cards a few times as we have been doing. Redo the missed ones as necessary and learn from them. 6. If there are any problems that you think you should add to the flash cards, do so at ANY time. 7 – 22. Copy the problem and rewrite in simplest form.

3 3

4 4 4 4 4 4

6 6 6 6 3 3

3 3 2 3 2 3 2 3

7. 8. 9. 10.

11. 12. ( ) 13. 14. 7 8

15. 7 8 16. 17. ( ) 18. 4 4

19. 4 4 20. ( ) 21. 22.

x x x x y y y y

z z z z z a a

a a b b b b c c

c c n n n n n

23 – 28. Generalize. Copy the problem and rewrite in simplest form.

5 2

23. 24.

25. 26.

27. ( ) 28. ( )

a a a a

c d c d

e f

x x x x

x x x x

x x

138

29 – 38. Copy the problem, expand as necessary, then simplify.

2 3 4 3 4 2 3 5 2

4 3 2 2 2 3 2 3 2

2 2 3 2 3 2

3 4 3 2 5 3

29. ( )( ) 30. ( )( ) 31. ( )(2 )

32. ( )( 3 ) 33. ( 6 )(6 ) 34. ( 5 )(5 )

35. 6 6 36. 5 5

37. (2 ) 38. (3 )

x y x y x y xy a b a b

a b ab c d c d c d c d

c d c d c d c d

x y x y

39. a) Look at problem 33. b) Are you being asked to add, subtract, multiply or divide in this problem? c) Look at problem 35. d) Are you being asked to add, subtract, multiply or divide in this problem? e) Explain in a couple of sentences how these two problems are different and

yet similar. Include HOW these problems are done differently also. DAY 3 TERMINOLOGY A TERM is a number, a variable, or the PRODUCT of a number and variable(s).

Each of the following is a single term: -6 x 2a b

3 44x y z 5(a+b)

Since each of the above has just ONE term, each is called a MONOMIAL. An expression with TWO terms is called a BINOMIAL. The following are binomials:

2 3 42 7 4325a x xy c d

An expression with THREE terms is called a TRINOMIAL. The following are trinomials: 2 4 5 4 3 2 212 45 2 16 40 25x x y y x y x y y

SIMILAR TERMS are terms with the exact same variable(s) and exact same exponent(s).

BIG RULE, HUGE RULE: You can ONLY add or subtract terms that are similar terms!

2 3 2 2 3 2 2 3 2 3 2 2

2 3 2

2 3 2

.1 4 ( 6 ) 10 [4 10 ] [ 6 ]

6 ( 5 )

6 5

EX a b a b a b a b a b a b a b a b

a b a b

a b a b

NOTE: These two terms cannot be combined They don’t have the same exponents!

139

When multiplying, multiply the coefficients, keep the same bases and add the exponents

4 2 3 4 3 2

4 3 2 1

7 3

.2 ( 3 )(5 ) ( 3 5)( )( )

( 15)( )( )

15

EX x y x y x x y y

x y

x y

When raising a power to a power, keep the base the same and multiply the exponents

3 4 3 4

12

.3 ( )EX y y

y

If a minus sign ( – ) is in front of parentheses, remove the parentheses and take the opposite of each term

3 3 3 3

3 3

3

.4 (2 5) ( 4 1) 2 5 4 1

[2 ] [ 4 ] [ 5 1]

5 4

EX x x x x x x x x

x x x x

x x

“A shortcut is often the quickest way to some place that you weren’t going.” - Anonymous Opportunities To Learn DAY 3 REVIEW BLOCK

1. 3

10 = ? Decimal = ? Percent

2. Solve and show that your solution does, in fact, check: 3 7 13a a

3. 214 =

140

4. Graph Of The Day

a) On its own pair of axes, neatly draw the graph of 2 4y x

b) State the slope c) State the coordinates of the y-intercept (as an ordered pair). d) State the coordinates of the x-intercept (as an ordered pair). 5. Go back and reread the definition of a TERM. State how many terms in each of the following:

2 2 2 3 4 5

2 2 3 2

) ) 3 5 ) 6

) ) 7 ) 3(2 )

a x b n n c a b c d

d a a e y y y f c d

6. Using a – f in exercise five above: a) Rewrite the ones that are monomials b) Rewrite the ones that are binomials c) Rewrite the ones that are trinomials 7. Practice the EXPONENTS flash cards a few times as we have been doing. Redo the missed ones as necessary and learn from them. 8. If there are any problems that you think you should add to the flash cards, do so at ANY time. 9 – 32. Copy and simplify. Show your work neatly. Use proper form.

4 3 2 4 3 2

4 3 2 4 3 2

2 2 2 2

2 2 2 2

2 2 2 2

9. 2 5 2 2 5 1

10. 5 2 5 2 2 5 1

11. (6 7) ( 5) 12. (7 4) (2 1)

13. (3 ) ( 4 )

14. (2 ) ( 4 2 )

y y y y y y y y

w w w w w w w w

a a a c c c

xy y x y xy y x y

ab a ab ab a ab

141

2 2 2 2

2 2 2 2

2 4 3 4 2 4 3 4

2 4 3 4 5 5 4 4

5 5 4 4 5 5 4 4

4 3 5 4 5 6

15. (3 ) ( 4 )

16. (2 ) ( 4 2 )

17. 18. 19. 20.

21. ( ) 22. ( ) 23. 24.

25. ( ) 26. ( ) 27. ( ) 28. ( )

29. ( 2 )(3 ) 30. ( 4 )(3

xy y x y xy y x y

ab a ab ab a ab

x x y y x x y y

x y z z a a

z z a a z a

x y xy xy x y

2 2

2 2

)

31. ( 3 5) ( 4 5)

32. (2 7) ( 5 3)

x x x x

x x x x

33. Horatio said that 2 2( 4) 16x x .

But Alger said that 2 2( 4) 8 16x x x .

To see who is correct, STOre 5 into X in your calculator and:

a) Evaluate 2( 4)x

b) Evaluate 2 16x

c) Evaluate 2 8 16x x

d) Who is correct and why?

e) Copy and complete: 2( 4)x

DAY 4 THE DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER ADDITION AND SUBTRACTION

Ex. 1 Rewrite using the distributive property (expand and simplify): 3(2 5)x

3(2 5) 3 2 3 5

6 15

x x

x

IN GENERAL: ( ) ( )a b c a b a c a b c a b a c

Ex. 2 Expand and Simplify: 22 ( 3 5)x x x This is called “a monomial X a trinomial”

2 2

1 2 1 1

3 2

2 ( 3 5) [ 2 ] [ 2 3 ] [ 2 5]

2 [ 6 ] 10

2 6 10

x x x x x x x x

x x x

x x x

“The severity of an itch is inversely proportional to the reach.” - Anonymous

142

Ex. 3 Expand and Simplify: (2 5)( 8)a a This is called “multiplying binomials”

2

2

(2 5)( 8) 2 ( 8) 5( 8)

[2 ] [2 8] [5 ] [5 8]

2 16 5 40

2 11 40

a a a a a

a a a a

a a a

a a

Alternative solution: FOIL First Outside Inside Last

2

2

(2 5)( 8) [2 ] [2 8] [5 ] [5 8]

2 16 5 40

2 11 40

F O I L

a a a a a a

a a a

a a

Ex. 4 Expand and Simplify: 2(3 8)n This is called “squaring a binomial”

2

2

2

(3 8) (3 8)(3 8) !

[3 3 ] [3 8] [ 8 3 ] [ 8 8]

9 24 24 64

9 48 64

n n n Now use FOIL

n n n n

n n n

n n

Please notice that when you square a binomial, your answer has THREE TERMS!!!

Ex. 5 Expand and Simplify: 2 5 3( 4 )x y This is called “cubing a monomial”

2 5 3 2 5 2 5 2 5

2 2 2 5 5 5

6 15

( 4 ) ( 4 )( 4 )( 4 )

[ 4 4 4] [ ] [ ]

64

x y x y x y x y

x x x y y y

x y

BIG QUESTION: Why didn’t we use the distributive property in example 5? “A friend is a present you give yourself.” - Robert Louis Stevenson

143


1. 1 7

2 8

2.

Point D is the midpoint of CE . (2 5) (5 7)CD x cm DE x cm

Find CD, DE, CE. 3. Find the slope and y-intercept of: 2x – 3y = 3. 4. Graph Of The Day

a) On its own pair of axes, neatly draw the graph of 3y

b) State the slope c) State the coordinates of the y-intercept (as an ordered pair). d) State the coordinates of the x-intercept (as an ordered pair). 5. What is the complete name of the property illustrated here:

( ) ( )a b c a b a c a b c a b a c

6. Using the method shown in Example 3, expand and simplify: (3 4)(2 1)y y .

Show each step. 7. What words does the acronym ‘FOIL’ represent?

8. Use ‘FOIL’ to expand and simplify: ( 8)( 2)x x .

9. Using the method shown in Example 4, expand and simplify: 2(4 3)n . Show each

step.

10. Using the method shown in Example 5, expand and simplify: 3 4 2( 5 )x y . Show

each step. 11. Practice the EXPONENTS flash cards a few times as we have been doing. Redo the missed ones as necessary and learn from them. 12. If there are any problems that you think you should add to the flash cards, do so at ANY time.

144

13 – 26. Copy the problem. Expand and simplify. Show any necessary steps using the proper form.

2 2 2 2 2 2

2 2

4 3 5 3

2 3 3 2 3 3

2 3 3 2 3 2

2

13. (4 3 ) 14. 2 (3 )

15. ( 6)( 8) 16. ( 5)( 4) 17. ( 7) 18. ( 5)

19. ( 2 ) 20. ( 3 ) 21. (3 2)( 5)

22. (2 5)( 4) 23. 3 ( 3 )

24. 4 ( 4 )

25. (4 3 ) 26. (2 5

x y x xy y x y x xy y

n n n n y y

a b ab n n

n n a b a a b ab b

a b a a b ab b

x y x y

2)

27. C.Y.O.P. a) Make up your own problem of squaring a binomial. b) Expand and simplify. c) Be prepared to share this problem tomorrow in class. 28. C.Y.O.P. a) Make up your own problem of squaring a monomial. b) Expand and simplify. c) Be prepared to share this problem tomorrow in class.

DAY 5 Opportunities To Learn DAY 5 REVIEW BLOCK

1. 7 7

2. How many sides does a decagon have?

145

3. a) Copy this problem onto your paper b) Point out where the error(s) occur c) Redo the problem correctly

2 6 3 4 3

12

4. Graph Of The Day

a) On its own pair of axes, neatly draw the graph of 2x

b) State the slope c) State the coordinates of the y-intercept (as an ordered pair). d) State the coordinates of the x-intercept (as an ordered pair). 5. Practice the EXPONENTS flash cards a few times as we have been doing. Redo the missed ones as necessary and learn from them. 6. If there are any problems that you think you should add to the flash cards, do so at ANY time. 7 – 36. Copy the problem and perform the indicated operations. Write your answer in simplest form. Show all necessary steps neatly and using the proper form.

3 3 4 4 3 3 4 4

3 3 4 4 2 3 4 3

2 3 4 3 2 3 4 3

2 3 4 3 4 4

3 4 2 5 2 2

2 3 2 2 3 2 3 2 2 3

7. 8. 9. 10.

11. ( ) 12. ( ) 13. 14.

15. 16. 17. ( ) 18. ( )

19. ( 3 )( ) 20. ( 4 )( )

21. ( ) 22. ( )

23. ( 2 3 ) 24. ( 2 3 )

25.

x x x x x x x x

x x y y y y

y y y y y y

a b ab a b ab

x y z xy z

xy x x y xy y x y x x y xy y

2 2

( 4)( 5) 26. ( 6)( 7)

27. ( 7) 28. ( 8)

x x x x

y y

146

2 2 2 2

2 2 2 2

5 2 3 4 3 3

2 2

29. (3 5 8) ( 6 1) 30. (2 7 5) ( 4)

31. (3 5 8) ( 6 1) 32. (2 7 5) ( 4)

33. ( 2 ) 34. ( 2 )

35. (3 7) 36. (2 5)

a a a a a a a a

a a a a a a a a

a b c a bc

n n

DAY 6 MORE RULES WITH EXPONENTS?! OH NO!!! Oh, yes. Your teacher will supply you with some examples. Opportunities To Learn DAY 6 REVIEW BLOCK

1. 225

2. A type of thinking when you observe something specific happen over and over again, and then you make a general conclusion about it, is called _________. 3. Copy the figure onto your paper neatly using a straightedge:

2 58j k m

Find the measure of each numbered angle.

4. Graph Of The Day

a) On its own pair of axes, neatly draw the graph of 2 3 3x y

b) State the slope c) State the coordinates of the y-intercept (as an ordered pair). d) State the coordinates of the x-intercept (as an ordered pair).

147

5. Write a few sentences to explain how you would simplify each of these two fractions. Then do them:

6 2

2 6) )

x xa b

x x

6. Make additional flash cards using the laws of exponents just reviewed today: Front of flash card Back of flash card

2 2

3 32

6 65

2 2

3 3 2

6 6 5

8 84

4 4

4 4

8 8 4

8 86

2 2

5 5

7 7 2

1

1

1

1

1

a aa

a a

a aa

a a

a aa

a a

a a

a a a

a a

a a a

a a

a a a

a aa

a a

a a

a a a

a aa

a a

a a

a a a

7. Practice the new EXPONENTS flash cards a couple of times until you feel that you know the rules for division pretty well.

148

8. Practice the old EXPONENTS flash cards a couple of times until you feel that you know them pretty well. 9 – 24. Copy the problem and simplify. No negative exponents in your answers.

3 4

3 4

6 2 2 8

2 8 6 2

3 12 12 4

12 4 3 12

3 5 4 2

2 8 10

9. 10. 11. 12.

13. 14. 15. 16.

17. 18. 19. 20.

16 20 16 621. 22. 23. 24.

2 4 6 27

a b c d

a b c d

e f g h

e f g h

i j k m

i j k m

a b c d

a b c d

DAY 7 DIVIDING A POLYNOMIAL BY A MONOMIAL EX. 1 Divide the polynomial by the monomial and simplify:

10 8 5 3 10 8 5 3

2 2 2 2 2

10 2 8 2 5 2 3 2

8 6 3

4 20 10 8 4 20 10 8

4 4 4 4 4

51 5 2

2

55 2

2

x x x x x x x x

x x x x x

x x x x

x x x x

149

EX. 2 Divide the polynomial by the monomial and simplify:

5 8 4 6 2 3 2 5 8 4 6 2 3 2

2 2 2 2 2

4 6 3 4

10 3 10 3

10 3 1

x y x y x y xy x y x y x y xy

xy xy xy xy xy

x y x y xy

***Don’t forget this 1! EX. 3 Divide the polynomial by the monomial and simplify:

12 9 8 6 4 3 12 9 8 6 4 3

4 3 4 3 4 3 4 3

8 6 4 3

18 12 6 18 12 6

6 6 6 6

3 2 1

a b a b a b a b a b a b

a b a b a b a b

a b a b

***Don’t forget this – 1! “One today is worth two tomorrows.” - Benjamin Franklin Opportunities To Learn DAY 7 REVIEW BLOCK

1. Simplify: 8

12

2. If the perimeter of this triangle is 22 in., find the length of each side. 3. COPY and complete:

A triangle with at least two sides equal in length is called _____.

150

4. Graph Of The Day

a) On its own pair of axes, neatly draw the graph of y x

b) State the slope c) State the coordinates of the y-intercept (as an ordered pair). d) State the coordinates of the x-intercept (as an ordered pair). 5. Practice the new EXPONENTS flash cards a couple of times until you feel that you know the rules for division pretty well. 6. Practice the old EXPONENTS flash cards a couple of times until you feel that you know them pretty well. 7 – 18. Divide the following polynomials by the monomials indicated and simplify. Show all steps using the proper form:

12 10 6 4 12 9 6 3

2 3

5 4 2 6 5 2

5 4 4 2 5 4 3 3 2

10 12 8 6 3 3 12 10 8 4 4 3

2 3 4 2

8 4 7 3

20 4 6 18 6 37. 8.

2 3

8 4 3 9 5 39. 10.

6 11 2 21 3 711. 12.

2 3

35 5 10 54 6 1213. 14.

5 6

36 10 415.

y y y y z z z z

y z

x x x y y y

x y

a b a b ab x y x y x y

ab xy

a b a b a b p q p q p q

a b p q

x y x y x

5 3 8 7 3 6 3 5

5 3 3 5

7 5 4 3 7 5 4 3

21 12 216.

4 3

24 18 6 24 18 617. 18.

6 3

y a b a b a b

x y a b

x x x x a a a a

19. C.Y.O.P. Create a problem similar to one like #7 through 18 above. Simplify the problem and be prepared to share this in class tomorrow. “There are times when silence has the loudest voice.” - Leroy Brownlow

151

DAY 8 DIVIDING A POLYNOMIAL BY A BINOMIAL

EX. 1 Divide and check:

2 3 5

1

x x

x

22

2

2

2

2

1 3 5 : :

1 3 5

( )

1 3 5

4 5 "

xx x x Think Divide the first term by the first term x

x

x

x x x Multiply using the distributive property

x x Subtract by taking the opposite of each term and then adding

x

x x x

x x Simplify and then

x brin

2

2

"

4

1 3 5

4: 4

4 5

( 4 4)

g down the next term

x

x x x

xx x Again divide the first term by the first term

x

x

x Next take the opposite and add

2

2

4

1 3 5

4 5

4 4

1

x

x x x

x x

x

x

This is the remainder

Therefore:

2 3 5 14

1 1

x xx

x x

152

To check a division problem: multiply the quotient by the divisor and add the remainder. The result should be the dividend (that is, what you divided “into”).

Check:

2

2

( 1)( 4) ( 1) 4 4 1

3 5 !!

x x x x x

x x it checks

EX. 2 Divide and check. Show each step using the proper form:

3 24 3 4

2

x x x

x

EX. 3 Divide and check. Show each step using the proper form:

2 9

4

y

y

Opportunities To Learn DAY 8 REVIEW BLOCK 1. Copy the following and fill in the blanks to complete the pattern: 1 , 2, 4 , 8 , 16, _____ , _____

2 ) 2 3 ) 2 3a b

3. Evaluate this expression using a calculator and round your final answer to the

nearest thousandth.

2 3

42.1 35

a ba b

ab b

4. Graph Of The Day

a) On its own pair of axes, neatly draw the graph of 4 0x y

b) State the slope c) State the coordinates of the y-intercept (as an ordered pair). d) State the coordinates of the x-intercept (as an ordered pair). 5. Practice the new EXPONENTS flash cards a couple of times until you feel that you know the rules for division pretty well. 6. Practice the old EXPONENTS flash cards a couple of times until you feel that you know them pretty well.

153

7 – 14. Divide each polynomial by the binomial. Show each step neatly using the proper form. Show that your answer does, in fact, check.

2 2

2 2

3 2 3 2

2 2

6 7 9 107. 8.

1 1

3 5 5 39. 10.

1 1

4 6 5 411. 12.

2 2

16 913. 14.

3 2

n n y y

n y

a a b b

a b

x x x y y y

x y

z t

z t

15 – 18. Copy the problem and simplify using the proper form:

3 3 3 3

3 3 3 2 5

15. 2 2 16. 2 ( 2 )

17. ( 2 ) 18. ( 4 )( )

b b b b

b x y xy

154

DAY 9 NEGATIVE EXPONENTS AND ZERO AS AN EXPONENT Investigate, investigate, investigate, … and DISCOVER! Copy the following table onto your paper. Use your graphing calculator to evaluate. You are to write the answers in two forms: decimal and fraction (using the MATH Frac key).

Number Evaluated as a DECIMAL

Evaluated as a FRACTION

12

13

14

15

16

Conjecture. Copy onto your paper and rewrite each as a fraction: 1 17 ; 8

Are you ready to generalize? Discuss with your partner: 1 ?x Write your answer

as an equation. Copy the following table onto your paper. Use your graphing calculator to evaluate. You are to write the answers in just one form: fraction (using the MATH Frac key).

Number Evaluated as a FRACTION

12

3

13

4

15

3

19

2

121

43

155

Conjecture. Copy onto your paper and rewrite each as a fraction. Verify with a calculator as necessary.


16

7

11

5

14

9

1

8

135

12

Are you ready to generalize? Discuss with your partner:

1

?n

d

Write your

answer as an equation. Now summarize in words and symbols.

Raising a number or expression to the negative one power means to take its

reciprocal, or

1n d

d n

and

1 1, 0, 0, 0x n d x

x

INVESTIGATE one more time! Copy the following table onto your paper. Use your graphing calculator to simplify.

Expression Simplified 04

0( 7)

03

7

041

16

156

Conjecture. Copy onto your paper and simplify. Verify with a calculator as necessary.

Expression Simplified 08

0( 3)

06

11

0

295.7

Are you ready to generalize? Discuss with your partner: 0 ?B

Before we jump to a wrong conclusion, we need to investigate one special case: 00

Enter 00 into your calculator and discuss the result with your partner.

Now summarize in words and symbols. Opportunities To Learn DAY 9 “What appears to be the end may really be REVIEW BLOCK a new beginning.” - Anonymous 1.

a) Name the hypotenuse b) Name the legs

2. Lines a and b are parallel. Lines a and c are perpendicular.

If the slope of line a is 2

3 , then find the slope for:

a) line b b) line c 3. First solve the equation for b. Then substitute to find the value for b.

1; 161 . . 7 .

2A bh A sq ft h ft

157

4. Graph Of The Day

a) On its own pair of axes, neatly draw the graph of 2 6 0y

b) State the slope c) State the coordinates of the y-intercept (as an ordered pair). d) State the coordinates of the x-intercept (as an ordered pair). 5. Maureen said that any number raised to the zero power is equal to one. Justin disagreed with Maureen. Who is correct? Explain why. 6. Copy and complete: Raising a number or expression to the negative one power is equivalent to _____. 7 – 26. Copy the problem onto your paper and simplify. Assume no variables have a value of zero. No calculators here.

0 0 1 1

1 1

0 0

01 1 0

1 1 1 1

12 4 2

3

7. 6 8. 12 9. 7 10. 11

1 211. 12. 13. ( 217) 14. ( 1053)

3 5

41 1515. 16. 17. 18.

29 17

2 519. 20. 21. 22.

3 4

323. 24.

c p

d q

x b a c

y a

x y a

z

1 1 1

4

1 225. 2 26. 4

3 5b c

27 – 32. Copy onto your paper and simplify. Show all necessary steps using the proper form.

2 2

2 3 2 2 2 2

1 1

27. ( 3 5) (3 5) 28. (2 5)( 3)

29. ( 8) 30. ( )( 3 ) (2 )( 5 )

31. 1 32. ( 1)

a a a a x x

y c d cd c d c d

158

DAY 10 Investigate, investigate, investigate, … and DISCOVER … AGAIN!! Copy the following table onto your paper. Use your graphing calculator to evaluate. You are to write the answers in one form: exact fraction (using the MATH Frac key).


22

23

24

25

Conjecture. Copy onto your paper and rewrite each as a fraction: 2 26 ; 8


as an equation. Copy the following table onto your paper. Use your graphing calculator to evaluate. You are to write the answers in one form: exact fraction (using the MATH Frac key).


32

33

34

Conjecture. Copy onto your paper and rewrite each as a fraction: 35

159


as an equation. Copy the following table onto your paper. Use your graphing calculator to evaluate. You are to write the answers in one form: fraction (using the MATH Frac key).


22

3

25

6

28

7

21

6

Conjecture. Copy onto your paper and rewrite each as a fraction:

2 23 7

;4 5

Are you ready to generalize? Discuss with your partner:

2

?n

d

Write your

answer as an equation. Discuss with your partner if you think a similar relationship will hold true for a fraction raised to the negative three power. Investigate and then make a conjecture. Now summarize in words and symbols. If a fraction is raised to a negative integer power, take the reciprocal of the fraction and then raise that answer to the power without the negative sign. Or, in symbols,

0, 0

k kn d

for n dd n

160

EX. 1 Rewrite without negative exponents in simplest form:

233x

y

2 23

3

3 3

1 1

3 3

2

6

3

3

3 3

3 3

9

x y

y x

y y

x x

y

x

y

x

EX. 2 Rewrite without negative exponents in simplest form:

34

5

2

3

a

bc

3 34 5

5 4

5 5 5

4 4 4

1 1 1 5 5 5

4 4 4

3 15 3 15 3 15

12 12 12

2 3

3 2

3 3 3

2 2 2

3 3 3

2 2 2

27 27 27

8 8 8

a bc

bc a

bc bc bc

a a a

b c

a

b c b c b cor or

a a a

Either answer is acceptable. Opportunities To Learn DAY 10 REVIEW BLOCK

1. Copy and simplify showing each step: 3 5 4

2. If 4 3x y , then ?x

y

161

3. SR = (6x – 7) in QR = 3 in.

If the perimeter of PQRS is 52 in.: a) Find the value for x b) Find the dimensions of the rectangle. 4. Graph Of The Day

a) On its own pair of axes, neatly draw the graph of 2 3 0x

b) State the slope c) State the coordinates of the y-intercept (as an ordered pair). d) State the coordinates of the x-intercept (as an ordered pair). 5 – 6. Copy the statement and complete: 5. A negative one exponent means to ____. 6. A negative two exponent first means to _____ and then raise that result to the _____ power. 7 – 26. Copy the problem onto your paper and rewrite the expression without negative exponents. Simplify as necessary.

1 1 2 2

2 2 3 3

2 2 3 32 4 4 5

3 3 2 3

2 7 9 87. 8. 9. 10.

5 9 4 5

11 12 3 111. 12. 13. 14.

8 7 4 2

15. 16. 17. 18.x a c d f

y b e gh

162

2 2 3 33 4 5 4 3

2 3 2 2

3 42 0 3 4 3 0

2 5

5 11 5 319. 20. 21. 22.

13 14 2 4

23. 7 24. 8 25. 26.

a b x c d a

c yz e b c

x yx y z a b c

x y

“What we all tend to complain about most in other people are those things we don’t like about ourselves.” - William Wharton DAY 11 Exponents – when do we need them? Answer: Scientific notation.

For numbers between 1000 and 1

1000, or 0.001, “regular” or standard notation is fine.

However, for extremely large numbers and extremely small numbers, we need a different notation – scientific notation. One example of an extremely large number would be the number of miles from the earth to the sun: 3,700,000,000 miles, which is read “three billion, seven hundred million miles.” In order to write numbers this large (and larger) and to be able to perform mathematical calculations on these numbers, it would be advantageous to have a better notation:

93,700,000,000 3.7 10

The number on the left is in standard notation, the number on the right is in scientific notation. One example of an extremely small number would be an Angstrom, which is used to

measure the wave length of light. An Angstrom is defined to be 1

100,000,000 of a

centimeter. Using scientific notation:

811 10

100,000,000

The number on the left is in standard notation, the number on the right is in scientific notation. As you can see, we use positive powers of ten to represent large numbers and negative powers of ten to represent tiny numbers.

163

Before we go any further, let’s discuss some powers of ten:

5

5 1 1 1 1 1 1 110 0.00001

10 10 10 10 10 10 100,000means or

Copy and complete the following table:

Power of 10 Meaning Whole number Fraction Decimal 310

210

110

010

110

210

310

All right now, so what exactly IS scientific notation? Scientific notation is a way of representing numbers as the product of a number between one and 10, and a power of ten. That is, scientific notation is in the form:

10pN where 1 10N and p is an integer.

NOTE: The number of digits in the number, N, is called the number of significant digits. Our goal is for you to be able to take a number that is in standard notation and to rewrite it in scientific notation and the reverse of that. EX. 1 Rewrite the following in scientific notation with the suggested number of significant digits: 45,200,000 a) three significant digits b) two significant digits c) one significant digit

a) three significant digits: 45,200,000 74.52 10

b) two significant digits: 45,200,000 74.5 10

c) one significant digit: 45,200,000 75 10

164

EX. 2 Rewrite the following in scientific notation with the suggested number of significant digits: 0.00000002351 a) three significant digits b) two significant digits c) one significant digit

a) three significant digits: 0.00000002351 82.35 10

b) two significant digits: 0.00000002351 82.4 10

c) one significant digit: 0.00000002351 82 10

EX. 3 Rewrite the following in standard form:

8 5) 4.73 10 ) 2.004 10a b

8) 4.73 10a means to multiply by 10,000,000 which will make the number larger by

“moving” the decimal point to the right 8 places, the same number as the number of zeros.

84.73 10 473,000,000

5) 2.004 10b means to multiply by 0.00001 which will make the number smaller

by “moving” the decimal point to the left 5 places. 52.004 10 0.00002004


1. Solve and show that your solution does check: 2 3 2

9 15

x x

2. Copy the problem and simplify (mentally if possible): 2 3 2 3

) )3 4 3 4

a b

3. SR = (6x – 7) cm RQ = 3 cm If the area of rectangle SPQR is 51 sq cm,:

a) Find the value for x b) Find the dimensions of the rectangle

165

4. Graph Of The Day

a) On its own pair of axes, neatly draw the graph of 4 0x y


5. A number that is written in scientific notation is in the form 10pN

a) What must be true about the values used for N? b) What kind of numbers must be used for p? 6. a) For numbers written in scientific notation, a positive exponent means to “move” the decimal point in which direction? b) For numbers written in scientific notation, a negative exponent means to “move” the decimal point in which direction? 7 – 18. Rewrite the number in scientific notation using the number of significant digits requested. 7. 48,000,000 using 2 significant digits 8. 9,600,000 using 2 significant digits 9. 0.0000613 using 3 significant digits 10. 0.00052 using 3 significant digits 11. 1,760,000,000 a) 3 significant digits b) 2 significant digits c) 4 significant digits 12. 21,400,000,000 a) 3 significant digits b) 2 significant digits c) 4 significant digits 13. 0.00000000041 a) 2 significant digits b) 1 significant digit c) 3 significant digits 14. 0.000000000013 a) 2 significant digits b) 1 significant digit c) 3 significant digits 15. –80,000,000,000,000 a) 2 significant digits b) 1 significant digit 16. –7,000,000,000,000 a) 2 significant digits b) 1 significant digit 17. –0.000000205 a) 3 significant digits b) 2 significant digits c) 1 significant digit 18. –0.0000000408 a) 3 significant digits b) 2 significant digits c) 1 significant digit

166

19 – 26. a) Copy the problem and rewrite in standard form b) State the number of significant digits

5 6 7 8

12 10 5 7

19. 4.7 10 20. 3.8 10 21. 4.3 10 22. 2.56 10

23. 6.02 10 24. 7.91 10 25. 2 10 26. 2.0 10

“Diplomacy is the art of letting someone else have your way.” - Anonymous DAY 12 Using Scientific Notation on the TI-83 On your calculator, press the <MODE> key. Have your calculator in NORMAL and FLOAT mode. Calculator notation

EX. 1 To enter 72.61 10 on the TI-83, type:

2.61 <2nd> <comma> (EE) <7> <ENTER> The calculator should show: 2.61 E 7 26,100,000 This is the calculator’s version of This is standard notation. scientific notation.

EX. 2 To enter 55.2 10 on the TI-83, type:

5.2 <2nd> <comma> (EE) < -5> <ENTER> The calculator should show: 5.2 E -5 5.2 E -5 This is the calculator’s version of The calculator will not put this in Scientific notation. standard notation unless the negative exponent is -3, -2, or -1.

167

EX. 3 <MODE> SCI FLOAT Enter into calculator: 56000 Press <ENTER>

The calculator should show: 5.6 E 4 which means 45.6 10

Enter into calculator: 0.0000471 Press <ENTER>

The calculator should show: 4.71 E -5 which means 54.71 10

EX. 4 <MODE> NORMAL FLOAT Simplify:

8

3 4

2.6 10

1.75 10 4.97 10

a) Answer in scientific notation with three significant digits b) Answer in standard notation with three significant digits

a) 142.99 10 b) 299,000,000,000,000

NOTE: In order to write an answer in standard notation with so many significant digits, you must FIRST write the answer in scientific notation with that number of significant digits.

EX. 5 Simplify: 255.17 10

a) Answer in scientific notation with three significant digits b) Answer in standard notation with three significant digits

a) 127.19 10 b) 7,190,000,000,000

EX. 6 The distance from New York City to Los Angeles is approximately 3000 miles. How many inches is that? a) Exact answer b) In scientific notation with three significant digits c) In standard notation with three significant digits a) 190,080,000 inches

b) 81.90 10 inches (approximately)

c) 190,000,000 inches (approximately)

168

EX. 7 Put each of the following in order from least to greatest:

a) 6 5 63.7 10 , 4.5 10 , 2.99 10

b) 7 8 85.2 10 , 6.2 10 , 7.04 10

Answers: a) 5 6 64.5 10 , 2.99 10 , 3.7 10

b) 8 8 76.2 10 , 7.04 10 , 5.2 10

Opportunities To Learn DAY 12 REVIEW BLOCK 1. Copy and complete: 0.75 = ? Fraction = ? Percent

2. Copy and simplify:

112 2) )1 12

2

a b

3. The ratio of the measures of two supplementary angles is 4:11. Find the measure of each. Show each step using the URESC method. 4. Graph Of The Day

a) On its own pair of axes, neatly draw the graph of 3 2 8x y


169

5. The following answers appeared on the screen of Max’s TI-83 calculator. Copy the calculator answer and write what that number is: i) as a number in scientific notation ii) as a number in standard notation a) 5.43 E 7 b) 2.041 E -6 c) -4.005 E 6 d) -2.30 E -5 6. The calculator displays one ‘E’ with its answers on the screen but has two E’s (EE) in gold on the calculator itself, above the comma key. Do you find that confusing? Also above the division symbol there is a lower case e that means something altogether different. Is this confusing to you? 7 – 16. Place your calculator into MODE SCI FLOAT. Perform the calculations. a) Write your answer in scientific notation using 3 significant digits. b) Write your answer in standard notation using 3 significant digits.

11 6 13 9

16 36

17 12 25

6 14 6 7

2 27 9

3 46 6

19 6 6 17

7. 4.91 10 2.07 10 8. 3.5 10 1.73 10

9. 3.74 10 10. 9.12 10

3 10 4 10 8 1011. 12.

9 10 3 10 3 10 7 10

13. 8.71 10 14. 3.65 10

4.261 10 7.052 1015. 16.

6.1 10 7.45 10 2.17 10 3.5 10

17. The distance between two cities is 2500 miles. How many inches is that: a) exactly b) in scientific notation with 3 significant digits c) in standard notation with 3 significant digits 18. The distance between two cities is 4500 miles. How many inches is that: a) exactly b) in scientific notation with 3 significant digits c) in standard notation with 3 significant digits

170

19. If your math teacher is exactly 75 years old, how many seconds has your math teacher lived: (use 1 year = 365 days) a) exactly b) in scientific notation with 3 significant digits c) in standard notation with 3 significant digits 20. If your English teacher is exactly 80 years old, how many seconds has your English teacher lived: (use 1 year = 365 days) a) exactly b) in scientific notation with 3 significant digits c) in standard notation with 3 significant digits 21 – 22. Put in order from least to greatest:

21. 8 7 73.8 10 , 4.01 10 , 3.9 10

22. 4 4 53 10 , 7 10 , 4 10

23 – 24. Put in order from largest to smallest:

23. 9 8 94.3 10 , 4.1 10 , 4.2 10

24. 5 4 59.7 10 , 5.2 10 , 4.0 10

25 – 30. Quick Review. Simplify (positive exponents only):

8

22 8 8

2

2 22 8

8

25. 26. 27.

2 928. 29. 30. :

5

xx x x

x

x x xx x Divide

x x

171

CHAPTER 5 REVIEW 1 – 6. Graph Of The Day a) On its own pair of axes, neatly draw the graph of the equation. b) State the slope. c) State the coordinates of the y-intercept (as an ordered pair). d) State the coordinates of the x-intercept (as an ordered pair).

1. 2 2. 4 3. 2 0

4. 2 3 0 5. 3 4 8 6. 2 5 5

y x x y

x y x y x y

7 – 8. How many terms in each of the following:

2 2 3 2 2

3 2 4 2

7. ) 5 ) 3 ) 4 1

8. ) 8 ) 4 9 ) 3(2 4)

a a b x yz c c d cd d

a y y b a z c x

9. A polynomial with one term is called a(n) _____. 10. A polynomial with two terms is called a(n) _____. 11 – 56. Copy the problem and simplify. (No zero or negative exponents in answers).

4 4 3 3 4 4 3 3

3 24 5 4 3 5 2

4 3 5 2 3 3 5 5

3 3 5 5 4 2 2 5 2

3 34 5

3 3 2 2

11. 12. 13. 14.

15. 16. 17. 18.

19. 20. 21. 6 6 22. 4 4

23. 6 6 24. 4 4 25. 4 2 26. 3

27. 2 28. 4

29. 2 5 4 1 30. 4 2 6 5 6

3

z z x x z z x x

y z y y z z

y y z z a a x x

a a x x c c d ab ab

a b xy

y y y y n n n n

2 2 2 2 2 2 2 2

4 34 4 3 3 4 3

4 4 3 3

3 2 5 3 2 2 4 5 2 4

1. 4 3 5 6 32. 7 7 8

33. ( ) 34. ( ) 35. 36.

37. 38.

39. 4 3 40. 4 3

x xy y x xy y a ab b a ab b

a a c c a c

a a c c

x y x x y y a b a a b b

172

2

2 22 3 4 5 2

10 12 4

2 3 12

2 7 8

8 3 3

12 9 3 16 12 4

3 4

8 4 7 3 5 3 7 3 6 2 4

5 3

41. 7 5 42. 8 10 43. 6

44. 4 45. 46.

47. 48. 49.

6 850. 51. 52.

4 10

15 18 12 40 16 3253. 54.

3 8

36 10 4 50 35 555. 56.

4

a a z z x

y x y z ab c

x y y

x y y

x a a

x a a

y y y x x x

y x

x y x y x y x y x y x y

x y

45x y

57 – 58. Divide and check:

2 23 5 4 657. 58.

1 3

y y x x

y x

59 – 76. Simplify. Rewrite with no zero or negative exponents.

0 10 1

1 1 1 1

1 1 2 2

2 23 3 4 5 2

2 3

59. 8 60. 27 61. 8 62. 27

7 8 2 263. 64. 65. 66.

3 3 5 9

1 3 9 567. 3 68. 2 69. 70.

4 8 4 7

2 371. 72. 73. 74.

3 4

1375.

c d a b

e c

2 24 2 3

3

1876.

15 11

a x y

bc z

173

77 – 82. Write in scientific notation using the number of significant digits requested: 77. 420,000,000 a) 2 significant digits b) 3 significant digits c) 1 significant digit 78. 580,000,000,000 a) 2 significant digits b) 3 significant digits c) 1 significant digit 79. 0.00000000007061 a) 2 significant digits b) 3 significant digits

c) 1 significant digit 80. 0.000000005807 a) 2 significant digits b) 3 significant digits

c) 1 significant digit 81. –17,480,000,000,000 a) 2 significant digits b) 3 significant digits

c) 1 significant digit 82. –304,900,000 a) 2 significant digits b) 3 significant digits

c) 1 significant digit 83 – 88. Use a calculator to compute your answer in scientific notation to three significant digits.

15 4 21 9

3617

7 18 11 5

25 16

83. 4.76 10 8.09 10 84. 2.311 10 7.16 10

2.6 103.872 1085. 86.

4.6 10 2.3 10 3.7 10 4.3 10

87. 2.7 10 88. 4.9 10

89. The distance between two cities is 8000 miles. How many inches is that? a) scientific notation to two significant digits b) standard notation to two significant digits 90. The distance between two cities is 6500 miles. How many inches is that? a) scientific notation to two significant digits b) standard notation to two significant digits 91. Your social studies teacher is exactly 63 years old. How many seconds is that? a) scientific notation to two significant digits b) standard notation to two significant digits 92. Your guidance counselor is exactly 100 years old. How many seconds is that? a) scientific notation to two significant digits b) standard notation to two significant digits 93 – 94. Write in standard form:

7 7 6

6 5 4

93. ) 3.06 10 ) 4.013 10 ) 2.9 10 ) 5.2 04

94. ) 7.124 10 ) 3.6 10 ) 9.07 10 ) 7.3 02

a b c d E

a b c d E

95 – 96. Put in order from least to greatest:

95. 6 7 67.6 10 , 4.95 10 , 4.3 10 96.

10 11 105.4 10 , 3.2 10 , 4.99 10

Documents

DAY 1 CHAPTER FIVE...135 Opportunities To Learn DAY 1 REVIEW BLOCK 1. a) classify ABC b) with a protractor, find m ABC c) classify CBD d) find m CBD 2. List at least four facts that