College Algebra Opener(s) 5/15/1It’s Batman Day, Beltane, Childhood Depression Awareness Day, Foster Care Day, Global Love Day, International Sunflower Guerilla Gardening Day, International Workers Day, Law Day, Lei Day, Loyalty Day, May Day, National Chocolate Parfait Day, National Purebred Dog Day, Save the Rhino Day, Silver Star Service Banner Day and World Asthma Day!!! Happy Birthday Wes Anderson, Tim McGraw, Theo Van Gogh, Kate Smith, Yiannis Ritsos, Joseph Heller, John Woo, Judy Collins, Terry Southern, Jack Paar, Glenn Ford and Calamity Jane!!!

Agenda

ENTICE ENGAGE EXTEND1. Opener (10)2. Pair Share (3)3. WC Share/Disc. (7)4. Math Talk (7?)5. Math Talk Reflection (3?)6. Disc. 1: Grades Check (5)7. Disc. 2: Vocabulary Chart Completion (20)8. Ind. Work 1: Text ?s, p. 176, #1-11 (20)9. Ind. Work 2: Box Construction (25)10. HW: Wksht. 3-5, p. 99-10011. Exit Pass (5?)

Essential Questions1. How do I (HDI) identify monotonicity, end behavior and

continuity?2. HDI determine a function’s inverse?3. HDI identify symmetry in shapes or figures?4. HDI find points of symmetry across points and lines?5. HDI describe graph transformations in families of functions?

Objective(s)1. Students will be able to (SWBAT) prove continuity.2. SWBAT look at a graph and determine monotonicity, continuity

and end behavior.3. SWBAT find a function’s inverse.4. SWBAT determine symmetry in shapes or figures.5. SWBAT predict points based on deductions of symmetry.6. SWBAT predict transformations based on differences between

parent and child functions.

5/1TODAY’S OPENER

Look at the displayed graph. Its function is y = |x2 – 4|1. Is the graph continuous

or discontinuous? Why?2. Is the graph continuous

at x = 2? Why?3. What is the end behavior

of the function?4. What is the

monotonicity of the function? THE LAST OPENER

Look at the displayed graph.1. What’s its equation?2. What’s it inverse?3. Verify the inverse using composition.4. Sketch a graph of the inverse based on

tables.5. Is the inverse a function? Why or why

not?6. Is the graph continuous or discontinuous?

Why?7. What is the end behavior and monotonicity of the function for x < 0?8. What is the end behavior and monotonicity of the function for x > 0? ELLs Accommodations

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

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

Standard(s) CCMS-HSF.BF.B.4a-c: Find inverse functions. (Write an expression for an

inverse, verify by composition that one function is the inverse of another and read values of an inverse function from a graph or a table.)

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: intercepts; intervals where the function is increasing, decreasing, positive or negative; relative maximums and minimums; and symmetries.)

CCMS-HSF.BF.B.3: Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx) and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Exit PassSolve this system of 3 equations. Use the diamond process OR a matrix equation.

−x − 5y − 5z = 24x − 5y + 4z = 19x + 5y − z = −20

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

The Last Exit PassWrite 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”

HOMEWORKSee the Hancock website or Google Classroom.

Math TalkWHAT’S MY FUNCTION?

The vertex of a downward facing parabola sits on the x-axis. Its axis of symmetry passes through the first and fourth quadrants.

Give an example of a function representing this parabola, in standard form.

Extra CreditPeriod 6 Period 7

MauroRosaMauro (2x)

CarlaEmily (2x)KevinAlyssa

Suggestions for Helping Mr. Keys to Improve Class1. Doing pairwork in pairs is optional: working on one’s own may substitute. In addition,

partners may be chosen rather than given…as 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. Keys’s 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 Class1. In each class, students will volunteer in some way…

a. Participate in the openerb. Ask a questionc. Perform a classroom management taskd. Do a demoe. Correct my mistake…correctlyf. Etc.

g.

Continuous

Discontinuous

RelativeMinimum

Extrema

Minimum

Absolute Extrema

Relative Maximum

Jump Discontinuity

AbsoluteMinimum

End Behavior

Infinite Discontinuity

Maximum

Absolute Maximum

Relative Extrema

Point Discontinuity

Point ofInflection

Monotonicity

Critical Points

FUNCTION BEHAVIOR VOCABULARYTerm Definition/Process Notation/Example

Discontinuous(Discontinuity)

A function or graph is discontinuous if you must lift your pencil in order to draw it.

Jump Discontinuity

A function or graph has jump discontinuity if it stops at a certain x-value then starts again at the same x-value but a different y-value.

Infinite Discontinuity

A function or graph has infinite discontinuity if the absolute value of the y-values becomes greater and greater as that graph gets closer and closer to a certain x-value.

Point Discontinuity

A function or graph has point discontinuity if there is just one x-value for which it is undefined but, for all others, the graph is smooth and continuous.

Continuous(Continuity)

A function or graph is continuous if at x = c (c being some number), the function:

1. Is defined (f(c) exists)2. Approaches the same y-

value on the left and right sides of x = c AND

3. That y-value is f(c).

End Behavior

The end behavior of a graph or function is what happens to the y-values as the absolute value of x becomes greater and greater.

Monotonic(Monotonicity)

A function or graph is monotonic on an interval Q if it is increasing on Q or decreasing on Q. Whether a graph is increasing or decreasing is always judged by viewing a graph from left to right.

CriticalPoints

Critical points are x-values where the function changes from increasing to decreasing or vice versa.

Extrema(Extremum)

Any absolute or relative maximum OR minimum points on a graph.

Maximum When a graph of a function is increasing to the left of x = c and decreasing to the right of x = c, then there is a maximum at x = c.

Minimum When a graph of a function is decreasing to the left of x = c and increasing to the right of x = c, then there is a maximum at x = c.

Point of Inflection

A point where a graph changes its curvature as illustrated to the right. The slope of the tangent line is undefined.

AbsoluteExtrema

Absolute extrema are the greatest values OR least values over the entire domain of a graph.

AbsoluteMaximum

The greatest value that a function assumes over its entire domain is called the absolute maximum.

AbsoluteMinimum

The least value that a function assumes over its entire domain is called the absolute minimum.

RelativeExtrema

Relative extrema are the greatest values OR least values over some interval of the domain of a graph.

RelativeMaximum

This is not the greatest value of a function on the domain, but it is the greatest y-value on some interval of the domain.

Relative Minimum

This is not the least value of a function on the domain, but it is the least y-value on some interval of the domain.

Coordinates of Symmetric Points

y-axis(Even

Function)

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

Symmetry withRespect to the Point:

Origin(Odd

Function)

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

Example: (1, 1) and (-1, -1) are on the graph.

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

Zero Matrix

Matrix

Matrix Sum

Elements

Scalar

Matrix Product

Matrix of nth order

Row Matrix

m x n Matrix

Square Matrix

Equal Matrices

Column Matrix

Dimensions

Matrix Difference

Scalar Product Matrix

Undefined Matrix

Additive Inverse Matrix

MATRIX VOCABULARY

Term Definition/Process Notation/ExampleMATRIX A rectangular array of terms called elements. It is

denoted with a capital letter.

A =

COLUMN MATRIX

A matrix with only 1 column.

B =

ROW MATRIX A matrix with only 1 row.

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

D =

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.

F =

EQUAL MATRICES

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

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.

4 5 68 9 10

4 8 7

4 5 6

4 2 08 10 127 5 3

4 2 0 … e1n

8 10 12 … e2n

7 5 3 … e3n

… … … … …en1 en2 en3 ... enn

0 0 0 0 0 0 0 0 0

4 28 3

4 28 3

information.

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

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

J - I = G

(j11 - i11) = (6 - 2) = 4(j12 - i12) = (6 - 4) = 2(j21 – i21) = (9 - 1) = 8(j22 – i22) = (9 - 6) = 3

ADDITIVE INVERSE MATRIX

The matrix –A which, when added to matrix A, will produce the zero or additive identity matrix. If G = then –G =

and it is called matrix G’s 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

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

G • L

SCALAR The number you multiply a matrix by. 5 • G

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

5G = 5 =

(g11 • 5) = (4 • 5) = 20(g12 • 5) = (2 • 5) = 10(g21 • 5) = (8 • 5) = 40(g22 • 5) = (3 • 5) = 15

4 28 3

2 41 6

6 69 9

6 69 9

2 41 6

4 28 3

4 28 3

2 41 6

10 2819 50

4 28 3

4 5 68 9 107 3 2

4 28 3

4 28 3

20 1040 15

4 28 3

-4 -2-8 - 3

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.

Range

Domain

Ordered Pair

Abscissa

Relation

Ordinate

Function

Vertical Line Test

Function Notation

Real #s

Function Evaluation

Independent Variable

Dependent Variable

D

R

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 NM A R SWhen 1 bases’s exponents are

Outside and inside

parentheses,

Multiply them.

When identical base’s exponents

are Beside each other AND

not inside parentheses,

Add them.

When an exponent

is Negative,

Reciprocalize (or flip) it.

When identical base’s exponents

are Over each other,

Subtract them.

EXAMPLES

RADICAL RULESRADICAL

VOCABULARY 3√27 3∗3∗3=27 3 is the index 27 is the radicand 3 is the root

+/- RADICALS 52√50 + 62√8 52√2∗25 + 62√2∗4 52√2 2√25 + 62√2 2√4 5*52√2 + 6*22√2 Add ONLY when you

have the same radicand

Subtract ONLY when you have the same radicand

Only + or – the coefficents!

¿ /÷ RADICALS 3√5 x * 3√25 x2

3√125 x32√252√100

= 2√ 25100 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

412 = 2√41

Numerator = radicand exponent

Denominator = radical sign index

RADICAL OPERATIONS5√32

Index? 5Radicand? 32Root? 22*2*2*2*2= 32

2√12 + 2√48 - 2√272√3∗4 + 2√3∗16 - 2√3∗92√4 √3 + 2√16 √3 - 2√9√32√3 + 4 √3 - 3√36 √3 - 3√33√3

(3 √12 )(2√21)3∗2 √12 √216 √12∗21 6 √3∗4∗3∗76∗2∗3 √736 √7

a4

b3

4 r 8

t 9

1614

8√816√3

4√9 z2

Your name Your period

Date Opener

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Extra Credit

Math Talk Reflection

Exit Pass

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Math Talk Reflection

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