College Algebra Opener(s) 2/27

t

2/27

Its Anosmia Awareness Day, World Spay Day, World NGO Day, National Strawberry Day, National Kahlua Day, No Brainer Day, Cigar Day and International Polar Bear Day!!! Happy Birthday Constantine the Great, Lotte Lehmann, Franchot Tone, Joan Bennett, Irwin Shaw, Dexter Gordon, Noah Emmerich, Henry Wadsworth Longfellow, Chelsea Clinton, Elizabeth Taylor, Josh Groban, Nancy Spungen, Gidon Kremer, Ralph Nader, Joanne Woodward, Lawrence Durrell and John Steinbeck!!!

Agenda

ENTICE

ENGAGE

EXTEND

1. Opener (7)

2. WC Share (5)

3. Math Talk (5)

4. Reflection (3)

5. Disc./Demo: HW ?s (10)

6. Lecture 1: Rules of Symmetry (20)

7. Models: Text ?s, p. 134, #s 6-12 (20)

8. HW Time: Text ?s, #14-26 even (15)

9. Exit Pass (5)

Essential Questions

1. How do I (HDI) identify symmetry in shapes or figures?

2. HDI find points of symmetry across points and lines?

Objective(s)

1. Students will be able to (SWBAT) determine the presence of symmetry in shapes or figures.

2. SWBAT find points of symmetry across the origin.

3. SWBAT find points of symmetry across the x or y axes.

4. SWBAT find points of symmetry across x=y or x=-y.

5. SWBAT deduce formulas for finding points of symmetry.

6. SWBAT draw complete figures based on predicted points of symmetry.

2/27

TODAYS OPENER

Put these 2 equations in your calculator. Then, in 2 different opener boxes, write the equations and sketch their graphs on their own mini-coordinate planes:

y = x3

y =

Again, using your calculator, look at the table of inputs/outputs for each equation and copy a portion of it in each mini-coordinate plane. Make sure you record both positive inputs and negative inputs. Now answer these questions:

Are the graphs symmetric around the origin?

Do you see a pattern in the inputs and outputs?

Solve the system by using a matrix equation.

2x y = 7

3x + y = 3

ELLs Accommodations

Talk to the text with all demos; provide 1-on-1 tutoring during individual work

DLs Accommodations

Talk to the text with all demos; provide 1-on-1 tutoring during individual work

Standard(s)

1. CCMS-HSF.IF.B.4: For a function that models a relationship between 2 quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. (Key features include symmetries.)

Exit Pass

Solve this system of 3 equations. Please use the diamond process.

x 5y 5z = 2

4x 5y + 4z = 19

x + 5y z = 20

DO YOU HAVE A QUESTION ABOUT ANYTHING WE DID IN CLASS TODAY?

The Last Exit Pass

Write the following relation as a table of values and as an equation:

the domain is all positive integers less than 10, the range is 3 times x, where x is a member of the domain

HOMEWORK

See the Hancock website or Google Classroom.

Math Talk

Whats x and y?

Find the values of x and y in the figure.

What theorems did you use to determine your answers?

Extra Credit

Period 6

Period 7

Rosa (3x)

Neo

Mauro (3x)

Bryan

Fabian

Emily (4x)

Luis

Carla

Esme

Suggestions for Helping Mr. Keys to Improve Class

1. Doing pairwork in pairs is optional: working on ones own may substitute. In addition, partners may be chosen rather than givenas long as work is accomplished.

2. Work on problems as a class group.

3. Always give partial credit on work.

4. If we meet 3 days a week, one day of HW less. If we meet 2 days a week, HW each day.

5. HW time should be provided in class.

6. Allow more choice in terms of method chosen to solve problems.

7. Offer a 2nd chance to redo or complete homework after the normal 10-minute HW ? time.

8. Presenting comprehensive notes for examination by Mr. Keys at the end of the quarter will earn extra credit.

9. Comprehensive notes will be allowed during quizzes.

10. Tests will reflect exactly what was practiced in class and on homework.

11. Mr. Keys should be positive.

12. Go more in depth on explaining rules.

13. Employ more repetition of problem types.

14. Offer an after-school study session once a week, based on a suggestion list.

15. Use Remind to let students know of Mr. Keyss early arrival.

16. Provide or use video supplements to the text.

17. With openers, models, examples, etc., check in after each iteration with, Is it enough?

18. Schedule dedicated review sessions.

19. Use games such as Kahoot and Quizlet.

20. Make time in class (and outside) for a computerized method of instruction.

21. Provide options for homework.

22. Allow test corrections for extra credit.

23. Make time and space in AcLab for peer tutoring. (Peer tutors earn extra credit.)

24. Students may select a math buddy for academic purposes if they choose.

25. Exit passes will be used more regularly.

26. Openers will be solved in a step-by-step process.

27. Upload class notes.

28. Learn vocabulary through definitions, examples, games and a key.

29. Mr. Keys will be more legible and specific in his corrections.

Suggestions for Helping Mr. Keys to Teach Class

1. In each class, students will volunteer in some way

a. Participate in the opener

b. Ask a question

c. Perform a classroom management task

d. Do a demo

e. Correct my mistakecorrectly

f. Etc.

Coordinates of Symmetric Points

y-axis

(Even Function)

(-a, -b) S if and only if

(a, b) S.

Example: (1, 1) and

(-1, -1) are on the graph.

Test: Substituting (a, b) and (-a, -b) into the equation produces equivalent equations.

Origin

(Odd Function)

Symmetry with

Respect to the Point:

(1, 1) and (-1, -1)

MATRIX VOCABULARY

Additive Inverse Matrix

Matrix Sum

Matrix Difference

Scalar

Scalar Product Matrix

Matrix of nth order

Zero Matrix

Undefined Matrix

Matrix Product

Row Matrix

Dimensions

Square Matrix

Column Matrix

Elements

Matrix

m x n Matrix

Equal Matrices

Term

Definition/Process

Notation/Example

MATRIX

A rectangular array of terms called elements. It is denoted with a capital letter.

4 5 6

8 9 10

A =

COLUMN MATRIX

A matrix with only 1 column.

4

8

7

B =

ROW MATRIX

A matrix with only 1 row.

4 5 6

C =

m x n MATRIX

A matrix with m rows and n columns, read m by n

Matrix A is a 2 x 3 matrix

DIMENSIONS

The number or rows and the number of columns in the matrix

Matrix A has dimensions of 2 rows and 3 columns.

SQUARE MATRIX

A matrix with the same number of rows and columns

4 2 0

8 10 12

7 5 3

D =

4 2 0 e1n

8 10 12 e2n

7 5 3 e3n

en1 en2 en3 ... enn

MATRIX of nth ORDER

A square matrix with n rows and n columns

E =

ZERO MATRIX

A matrix in which all elements equal 0. Also known as the additive identity matrix.

0 0 0

0 0 0

0 0 0

F =

EQUAL MATRICES

2 matrices that have the same dimensions AND are identical, element by element.

4 2

8 3

4 2

8 3

If G = and H = ,

then G and H are equal matrices.

ELEMENTS

The terms arranged in a rectangular array within a matrix. They are arranged in rows and columns, enclosed by brackets and represented using double subscript notation, where the first subscript refers to the row and the second refers to the column. Elements can be numbers OR information.

Matrix A has 6 elements: {4, 5, 6, 8, 9, 10}.

4 is element a11, 5 is element a12, 6 is element a13, 8 is element a21, 9 is element a22 and 10 is element a23.

MATRIX SUM

The sum of 2 matrices exists only if the 2 matrices have the same dimensions. Elements with the same subscript are added together to create a 3rd element with the same subscript in a 3rd matrix. In other words, if matrix A with elements aij is added to matrix B with elements bij, then a 3rd matrix, C, is formed consisting of elements aij + bij = cij.

G + I = J

6 6

9 9

2 4

1 6

4 2

8 3

G + I = J

(g11 + i11) = (4 + 2) = 6

(g12 + i12) = (2 + 4) = 6

(g21 + i21) = (8 + 1) = 9

(g22 + i22) = (1 + 6) = 9

MATRIX DIFFERENCE

The difference of 2 matrices exists only if the 2 matrices have the same dimensions. Elements with the same subscript are subtracted from each other to create a 3rd element with the same subscript in a 3rd matrix. In other words, if matrix B with elements bij is subtracted from matrix A with elements aij, then a 3rd matrix, C, is formed consisting of elements aij bij = cij.

J I = G

4 2

8 3

2 4

1 6

6 6

9 9

J - I = G

(j11 - i11) = (6 - 2) = 4

(j12 - i12) = (6 - 4) = 2

(j21 i21) = (9 - 1) = 8

(j22 i22) = (9 - 6) = 3

4 2

8 3

ADDITIVE INVERSE MATRIX

The matrix A which, when added to matrix A, will produce the zero or additive identity matrix.

-4 -2

-8 - 3

If G = then G =

and it is called matrix Gs additive inverse

MATRIX PRODUCT

The product of 2 matrices exists only if the number of columns in the 1st matrix is identical to the number of rows in the 2nd matrix. Consider a matrix A of dimensions m x n and a matrix B of dimensions n x o. The product is found by multiplying each element in a row of A with a corresponding element in EACH column of B. The result is a 3rd matrix, C, of dimensions m x o.

G I = K

10 28

19 50

2 4

1 6

4 2

8 3

G I = K

(g11 i11) + (g12 i21) = (4 2) + (2 1) = 10

(g11 i12) + (g12 i22) = (4 4) + (2 6) = 28

(g21 i11) + (g22 i21) = (8 2) + (3 1) = 19

(g21 i12) + (g22 i22) = (8 4) + (3 6) = 50

UNDEFINED MATRIX

If the number of columns in matrix A does NOT match the number of rows in matrix B, then the product of A and B is undefined. In other words, if matrix A has dimensions m x n and matrix B has dimensions o x p, AB is undefined or impossible.

G L = Undefined

4 5 6

8 9 10

7 3 2

4 2

8 3

G L

SCALAR

The number you multiply a matrix by.

5 G

4 2

8 3

5 G = 5

MATRIX SCALAR PRODUCT

The product of a scalar k and an m x n matrix A is an m x n matrix denoted by kA. Each element of kA equals k times the corresponding element of A.

5 G = 5G

20 10

40 15

4 2

8 3

5G = 5 =

(g11 5) = (4 5) = 20

(g12 5) = (2 5) = 10

(g21 5) = (8 5) = 40

(g22 5) = (3 5) = 15

Geometric Figure Matrices

The Graph

The Coordinates written in Matrices

1. f(x) [f of x] is interpreted as the value of f at x.

2. The representation for range.

3. Drawing an up-and-down line through a graph to prove its function status.

4. The first element of an ordered pair.

5. Plugging a domain element into a function in order to determine the corresponding range element.

6. The set of all #s that can be represented as either a finite or infinite decimal.

7. Pairing the elements of one set with another set.

R

D

Dependent Variable

Function Evaluation

Relation

Function Notation

Vertical Line Test

Real #s

Independent Variable

Function

Ordered Pair

Ordinate

Range

Abscissa

Domain

8. The second element of an ordered pair.

9. The representation for domain.

10. Y

11. A special type of relation in which each domain element is paired with exactly one range element.

12. The set of all ordinates.

13. The set of all abscissas.

14. A pairing of an abscissa with an ordinate contained within parentheses and separated by a comma.

15. X

Exponent Rules

B

O

Y

O

N

M

A

R

S

When 1 basess exponents are Outside and inside parentheses,

Multiply them.

When identical bases exponents are Beside each other AND not inside parentheses,

Add them.

When an exponent

is Negative,

Reciprocalize (or flip) it.

When identical bases exponents are Over

each other,

Subtract them.

EXAMPLES

RADICAL RULES

RADICAL VOCABULARY

3 is the index

27 is the radicand

3 is the root

+/- RADICALS

5 + 6

5 + 6

5 + 6

5*5 + 6*2

Add ONLY when you have the same radicand

Subtract ONLY when you have the same radicand

Only + or the coefficents!

RADICALS

*

=

If index is the same,

put entire product under 1 radical sign

If index is the same, put entire division under 1 radical sign

Or, if everything is already under 1 radical sign, split in 2!

RATIONAL EXPONENTS

=

Numerator = radicand exponent

Denominator = radical sign index

RADICAL OPERATIONS

Index? 5

Radicand? 32

Root? 2

2*2*2*2*2= 32

+ -

+ -

+ -

+ -

-

( )(2)

Your name

Your period

Date

Opener

Question

Answer

Extra Credit

Math Talk Reflection

Exit Pass

Question

Answer

Date

Opener

Question

Answer

Extra Credit

Math Talk Reflection

Exit Pass

Question

Answer

Date

Opener

Question

Answer

Extra Credit

Math Talk Reflection

Exit Pass

Question

Answer

Date

Opener

Question

Answer

Extra Credit

Math Talk Reflection

Exit Pass

Question

Answer

Date

Opener

Question

Answer

Extra Credit

Math Talk Reflection

Exit Pass

Question

Answer

Date

Opener

Question

Answer

Extra Credit

Math Talk Reflection

Exit Pass

Question

Answer

Date

Opener

Question

Answer

Extra Credit

Math Talk Reflection

Exit Pass

Question

Answer

Date

Opener

Question

Answer

Extra Credit

Math Talk Reflection

Exit Pass

Question

Answer

Date

Opener

Question

Answer

Extra Credit

Math Talk Reflection

Exit Pass

Question

Answer

Kuta Software - Infinite Algebra 2

Determinants of 22 Matrices

Evaluate the determinant of each matrix.

Name

Date Period

0

6

4

2

6

6

0

6

1) 2)

1

1

1

4

0

6

4

5

3) 4)

2 0 1 2 K u t a S o ft w a r e L L C. A l l r i g ht s r e s e r v e d . M a d e w i t h I n f i n i t e A lg e b r a 2 . Worksheet by Kuta Software LLC

0 1 5 3

5)

6 6

6)

6 6

Evaluate each determinant.

5 3

7)

4 2

8)

9 9

7 10

1 8

9)

5 0

10)

8 6

10 9

11)

0 6

8 0

12)

10 9

7 3

13)

5 0

2 10

14)

2 2

7 7

15) Evaluate:

1 2

3 4

+

5 2

2 6

16) Give an example of a 22 matrix whose

determinant is 13.

0 1 5 3

5) 6 6 6) 6 6

Evaluate each determinant.

(Kuta Software - Infinite Algebra 2Determinants of 22 MatricesEvaluate the determinant of each matrix.) (Name ) (Date Period ) (06) (42) (66) (06) (1)) (2)) (11) (14) (06) (45) (3)) (4))

( 2 0 1 2 K u t a S o ft w a r e L L C. A l l r i g ht s r e s e r v e d . M a d e w i t h I n f i n i t e A lg e b r a 2 .) (Worksheet by Kuta Software LLC)

5 3

7) 4 2 8)

9 9

7 10

1 8

9) 5 0 10)

8 6

10 9

11)

0 6

8 0 12)

10 9

7 3

13)

5 0

2 10 14)

2 2

7 7

15) Evaluate:

1 2

3 4 +

5 2

2 6

16) Give an example of a 22 matrix whose determinant is 13.

3 24

(Kuta Software - Infinite Algebra 2Determinants of 22 MatricesEvaluate the determinant of each matrix.) (Name ) (Date Period ) (06) (42) (6 06 6) (1)) (2)) (24) (36) (11) (14) (06) (45) (3)) (4))

0 1

5) 6 6

6

5 3

6) 6 6

12

Evaluate each determinant.

5 3

7) 4 2

9 9

8) 7 10

22 27

1 8

9) 5 0

10)

8 6

10 9

40 12

11)

0 6

8 0

12)

10 9

7 3

48 33

13)

5 0

2 10

14)

2 2

7 7

50 0

15) Evaluate:

1 2

3 4 +

5 2

2 6

16) Give an example of a 22 matrix whose determinant is 13.

4 13

32 Many answers. Ex: 1 5

Create your own worksheets like this one with Infinite Algebra 2. Free trial available at KutaSoftware.com

Kuta Software - Infinite Algebra 2

Determinants of 33 Matrices

Evaluate the determinant of each matrix.

Name

Date Period

3

1) 3

2

1

1

2

3

0

2

1

3

1

2)

3 2 3

3 0 3

2 0 12 K u t a S of t w a r e L L C . A l l r i g h t s r e s e r v e d . M a d e w i t h I n f i n i t e A l g ebr a 2 . Worksheet by Kuta Software LLC

0

a

b

12) What value o

11) 0 c d

2 0 0

0 x y

6 x 1

Evaluate each determinant.

5

3

3

6

6

1

3) 4 5 1 4) 3 5 2

5 3 0

4 3 3

6

2

1

2

5

4

5) 5 4 5 6) 0 3 5

3 3 1

5 5 6

3

4

5

6

5

3

7) 4 6 3 8) 5 4 2

1 4 3

1 4 5

1

8

9

5

5

5

9) 4 12 7 10) 8 9 3

10 3 2

8 5 9

f x makes the determinant 4?

4

0

1

Evaluate each determinant.

5

3

3

6

6

1

3)

4

5

1

4)

3

5

2

5

3

0

4

3

3

6

2

1

2

5

4

5)

5

4

5

6)

0

3

5

3

3

1

5

5

6

3

4

5

6

5

3

7)

4

6

3

8)

5

4

2

1

4

3

1

4

5

1

8

9

5

5

5

9)

4

12

7

10)

8

9

3

10

3

2

8

5

9

(0ab 12) What value o11)0cd2000xy6x1)f x makes the determinant 4?

(Kuta Software - Infinite Algebra 2Determinants of 33 MatricesEvaluate the determinant of each matrix.) (Name ) (Date Period ) (31) 3) (21) (12) (30) (21) (31) (2)) (3 2 3) (3 0 3)

4 0 1

( 2 0 12 K u t a S of t w a r e L L C . A l l r i g h t s r e s e r v e d . M a d e w i t h I n f i n i t e A l g ebr a 2 .) (Worksheet by Kuta Software LLC)

12

24

Evaluate each determinant.

5

3

3

6

6

1

3)

4

5

1

4)

3

5

2

5

3

0

4

3

3

39 103

6

2

1

2

5

4

5)

5

4

5

6)

0

3

5

3

3

1

5

5

6

161

51

3 4 5

7) 4 6 3

1 4 3

6 5 3

8) 5 4 2

1 4 5

200

139

1 8 9

9) 4 12 7

10 3 2

10)

5 5 5

8 9 3

8 5 9

647

800

0 a b

12) What value of x makes the determinant 4?

11)

0

c

d

2

0

0

0

x

y

6

x

1

4 0 1

0

2

Create your own worksheets like this one with Infinite Algebra 2. Free trial available at KutaSoftware.com