UNIT ONE MM1A1. Students will explore and interpret the characteristics of functions,

using graphs, tables, and simple algebraic techniques. a. Represent functions using function notation.

b. Graph the basic functions f(x) = x(n), where n = 1 to 3, f(x) = √x, f(x) = |x|, and f(x) = 1/x.c. Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the x- and y-axes.d. Investigate and explain the characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior.e. Relate to a given context the characteristics of a function, and use graphs and tables to investigate its behavior.f. Recognize sequences as functions with domains that are whole numbers.g. Explore rates of change, comparing constant rates of change (i.e., slope) versus variable rates of change. Compare rates of change of linear, quadratic, square root, and other function families.

MM1G2. Students will understand and use the language of mathematical argument and justification. a. Use conjecture, inductive reasoning, deductive reasoning, counterexamples,

and indirect proof as appropriate.b. Understand and use the relationships among a statement and its converse, inverse, and contrapositive.

UNIT TWO

MM1A2 Students will simplify and operate with radical expressions, polynomials, and rational expressions. a. Simplify algebraic and numeric expressions involving square root.

b. Perform operations with square roots.c. Add, subtract, multiply, and divide polynomials.d. Expand binomials using the Binomial Theoreme. Add, subtract, multiply, and divide rational expressions.f. Factor expressions by greatest common factor, grouping, trial and error, and special products limited to the formulas below.

(x + y)2 = x2 + 2xy + y2 (x - y)2 = x2 - 2xy + y2 (x + y)(x - y) = x2 - y2 (x + a)(x + b) = x2 + (a + b)x + ab (x + y)3 = x3 + 3 x2y + 3xy2 + y3 (x - y) 3 = x3 - 3 x2y + 3xy2 – y3

g. Use area and volume models for polynomial arithmetic. MM1A3. Students will solve simple equations.

a. Solve quadratic equations in the form ax² + bx + c = 0, where a = 1, by using factorization and finding square roots where applicable.

UNIT THREE

MM1G3. Students will discover, prove, and apply properties of triangles, quadrilaterals, and other polygons. a. Determine the sum of interior and exterior angles in a

polygon.b. Understand and use the triangle inequality, the side-angle inequality, and the exterior-angle inequality.c. Understand and use congruence postulates and theorems for triangles (SSS, SAS, ASA, AAS, HL).d. Understand, use, and prove properties of and relationships among special quadrilaterals: parallelogram, rectangle, rhombus, square, trapezoid, and kite. e. Find and use points of concurrency in triangles: incenter, orthocenter, circumcenter, and centroid.

•The slope of a line passing through point (1,3) is (-3).•Write the rule (or equation) of this line.

Start-up Activity

m = y2 – y1 x2 – x1 -3 = y – 3

x - 1y – 3 = (-3)(x – 1)y – 3 = -3x + 3

y = - 3x + 6y = -3(x – 2)

True for all Real Numbers a, b, c

Important Properties

Addition MultiplicationCommutative a + b = b + a ab = ba

Associative (a+b) + c = a+(b+c)

(ab)c =a(bc)

Distributive a(b+c) = ab + ac and (b+c)a = ba + ca

Adding & Subtracting Expressions

Adding Expressions Subtracting expressions

Remove the parentheses

Use Commutative & Associative Properties to Rearrange and Group “like terms”

Simplify by combining “like terms”

Change the subtracted expression to its opposite

Add the expressions

(3x + 4) + (2x - 1)

(4d – 2) – (5d – 3)

(4d – 2) + (-5d + 3)4d – 2 + 3 – 5d4d – 5d – 2 + 3- d + 13x + 4 + 2x

– 13x + 2x +4 -1 5x +3

Multiplying & Dividing ExpressionsMultiplying Expressions Dividing Expressions ab = ba (ab)c = a(bc) a(b + c) = ab + ac

a bc

a b a bc c c

a b a bc c c

a b a bc c c

Dividing by Fractions - NASCAR Distance = Rate * Time Distance /Rate = Rate *Time/Rate Time = Distance/Rate Time = 1 Miles/ x Miles/Minute Time = 1 Miles * Minute/ x Miles f(x) = (1/x ) Minutes

Ticket Out The Door 9/17

Write a Paragraph containing at least 3 complete sentences:

The most interesting thing I learned in Math since school started. What was it about this topic that you liked most?

What I hope to learn more about this week.

Typical Binomial Squares

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 – 2ab + b2

Prove using F.O.I.L. (a + b) (a + b) = a2 + ab + ab + b2

= a2 + 2ab + b2

Sum and Difference Pattern (Conjugates) (a + b) (a - b) = a2 - ab + ab - b2

= a2 – b2

Binomial Theorem & Pascal’s Triangle

(a + b)0 = 1 (a + b)1 = 1a + 1b (a + b)2 = 1a2 + 2ab + 1b2

(a + b)3 = 1a3 + 3a2b +3ab2 + 1b3

11 1

1 2 11 3 3 1

Solving Polynomial Equations

In Factored Form Factoring and the Zero-Product Property If ab = 0, then:

a = 0, or b = 0, or Both a and b = 0

If (x + 1) (x – 2) = 0, then (x + 1) = 0, or (x - 2) = 0, or Both are zero

Solving Polynomial Equations(cont.) Factor x2 + bx + c

Factor ax2 + bx + c Factor Special Products

a2 – b2

( a + b) 2

(a - b) 2

Factor Polynomials Completely Written as a product of unfactorable

polynomials with integer coefficients

Mid-Term Assessment1. How do you convert Miles per Hour to Miles per Minute?

2. In the NASCAR problem, it was discovered that the Rational Function, f(x) = 1/x, was a good fit to reflect the relationship described in the problem. Identify f(x), 1, and x in this problem.

3. In what units was time to be measured in #2, above?

4. After an accident on the track which caused a 5-minute delay, (a) reflect this delay in a new f(x) relationship. (b) What impact will this have on the original graph?

5. The problem then changed the venue to Daytona, which has a 2.5 mile track. (a) Write a new f(x) which reflects this change. (b) What impact does this distance change have on the original graph?

6. In the Tiling Task, what is the relationship between Mario’s Figure No. and (a) number of rows? (b) Number of tiles in each figure?

7. Mario has 5,000 tiles and wants to know how many rows he needs to use them. How should he proceed?

8. How many rows will he use to place the maximum number of available tiles?

9. How were Triangular Numbers defined in the problem?

10. What do you get when you add two Triangular Numbers?

NASCAR Task

DAY 1MPH MPM Minutes

Dale Earnhardt, Jr. 180 3.00 0.33Mark Martin 60 1.00 1.00Jeff Gordon 240 4.00 0.25Kasey Kahne 120 2.00 0.50Tony Stewart 30 0.50 2.00Jimmie Johnson 90 1.50 0.67Jeff Burton 150 2.50 0.40Matt Kenseth 15 0.25 4.00

Mario’s Tiling Vision

Figure No. 1 2 3 4 5 6

No. Rows 1 3 5 7 9 11

No. Tiles 1 4 9 16 25 36

Latasha’s Tiling Vision

Figure No. 1 2 3 4 5 6

No. Rows 2 4 6 8 10 12

No. Tiles 2 6 12 20 30 42

Triangular Numbers

. ...

...…

...…….

...…….…..

Fig 1 2 3 4 5T.N. 1 3 6 10 15

Sum of the Numbers 1 through n is: ( 1)2

n n

YE OLDE VILLAGE SHOPPE

Define the Variables Draw a Picture Define Relationships Construct a Table Graph the Results Draw Conclusions

xww

2 802 80

802

402

( ) (40 / 2)

w xw x

xw

xw

A x x x

Ladder Length Learning Task

Warm-up

Simplify using the Distributive Property.

1. 4(x – 3) 2. -2(x + 5) 3. x(x + 10) 4. -x(x – 7)

Warm-up

Simplify using the Distributive Property. 1. 4(x – 3) A) 4x – 3 B) 4x + 12 C) x – 12 D) 4x – 7 E) 4x -

12 2. -2(x + 5) A) -2x + 3 B) -2x – 10 C) 2x – 10 D) -2x +

5 E) x - 10 3. x(x + 10) A) 2x + 10x B) 2x + 10x C) x2 + 10 D) x2 +

10x E) 10x2

4. -x(x – 7) A) -7B) 7x2 C) -x2 + 7x D) x2 - 7x E) 7x

Number Magic

Think of a number less than 10 Double it Add 10 Divide it in half Subtract the 1st number you thought of

from the answer Your final answer is 5!

(or you did something wrong!!!)

I’ve Got Your Number

Pattern 1: (x+a)(x+b) = x2 + (a + b)x + ab

(72)(76) A. (70 + 1)(75 + 1) B. (70 +2)(70 + 6) C. (68 +3)(70 + 6) D.

(70 + 2)(72 + 3)  

(31)(39) A. (30 + 1)(30 + 9) B. (27 + 5)(35 + 3) C. (29 + 4)(30 + 9) D.

(30 + 1)(32 + 5)  

(52)(48) A. (43 + 8)(40 + 8) B. (50 +2)(50 - 2)C. (43 + 9)(43 + 6) D. (40 +

12)(40 + 8)  

I’ve Got Your NumberPattern 1 HW (x+a)(x+b) = x2 + (a+b)x + ab

1. (x + 3)(x + 4) x2 + 7x + 12

2. (x + 5)(x + 2) x2 + 7x + 10

3. (x + 9)(x + 1) x2 + 10x + 9

4. (x + 7)(x + 6) x2 +13x + 42

I’ve Got Your NumberPattern 1 HW #5

1

x

2x

2

(x+2)(x+1) = x2 + 3x + 2x2 2x

1x

I’ve Got Your NumberProblem #1, a. b. c. Case 1: a

positive, b positive: Figure A

Case 2: a positive, b negative: Figure D

Case 3: a negative, b positive: Figure B

Case 4: a negative, b negative Figure C

Problem #1, a. b. c. Case 1: a positive,

b positive: Figure A

Case 2: a positive, b negative: Figure D

Case 3: a negative, b positive: Figure B

Case 4: a negative, b negative Figure C

I’ve Got Your Number

I’ve Got Your Number #2.

Use Pattern 1 to calculate each of the following products.  (a) (52)(57) =

(50 + 2 ) (50 + 7) = 502 + (2 + 7) (50) + (2)(7) = 2500 + 450 + 14 = 2964

(b) (16)(13) = (10 + 6)(10 + 3) = 102 + (6 + 3)(10) + (6)(3) = 100 + 90 + 18 = 208

(c) (48)(42) = (40 + 8)(40 + 2) = 402 + (8 + 2)(40) + (8)(2) = 1600 + 400 + 16 = 2016

(d) (72)(75) = (70 + 2)(70 + 5) = 702 + (2 + 5)(70) + (2)(5) = 4900 + 490 + 10 = 5400

I’ve Got Your Number #3.

(a) (34)(36) = 1224

(b) (63)(67) = 4221

(c) (81)(89) = 7209

(d) (95)(95) =

9025

Verify with Pattern 1 = 900 + 10(30) + 24 = 1224

Verify with Pattern 1 = 3600 + 10(60) + 21 = 4221

Verify with Pattern 1 = 6400 + 10(80) + 9 = 7209

Verify with Pattern 1 = 8100 + 10(90) + 25 = 9025

I’ve Got Your Number #3a.-f. To represent two-digit numbers with the same

ten’s digit start by using n to represent the ten’s digit. So, n is 3 for part (a) What is n for parts (b), (c), and (d)?

Next, represent the first two-digit number as 10 n + a and the second one as 10 n + b. In part (a):

(32)(38) = (30 + 2)(30 + 8) = (10n + a)( 10n + b) for n = 3, a = 2, and b = 8.

Now figure (b), (c), & (d)

I’ve Got Your Number #3h

(32)(38) = (30 + 2)(30 + 8) = (10n + a)( 10n + b) for n = 3, a = 2, and b = 8

Since a + b = 10, = 100n2 + (10)(10n) + ab = 100n2 + 100n + ab = 100(n2+ n) + ab

Let k = (n2 + n) = 100k + ab

Therefore, (32)(38) = 100(3)(4) + 16 = 1216

n2 + n = n(n+1)

I’ve Got Your Number #4

How much Bigger? How much bigger is the

area of a square if you add 4 to its length and width? (x+4)(x+4) = x2 + 8x + 16

How much bigger if you add y to length and width? (x + y)(x + y) x2 + 2xy + y2

Pattern 2: Square of a Sum

x2

x

x

4

4

x + 4

x + 4

I’ve Got Your Number #5

1. 3022 = (300 + 2)2 = (300)2 + 2(300)(2) + 22 = 90000 + 1200 + 4 =

912042. 542 =

(50 + 4)2 = (50)2 + 2(50)(4) + (4)2 = 2500 + 400 + 16 = 29163.   852 =

(80 + 5)2 = (80)2 + (80)(5) + (5)2 = 6400 + 800 + 25 = 7225

4. 2.12= (2 + 0.1)2 = (2)2 + 2(2)(0.1) + (0.1)2 = 4 + 0.4 + 0.01 = 4.41

 

I’ve Got Your Number #6

What is the Volume? If Length = Width = Height

= x How much larger if sides

increase by 4? (x + 4)(x2 + 8x + 16) = x3 + 12x2 + 48x + 64

How much larger is sides increase by y? (x+y)3 = x3+3x2y+3xy2+y3

Pattern 3: Cube of a Sum

xx

x x3

I’ve Got Your Number #6e

1. 113 = (10 + 1)3 = 103 + 3(10)2(1) + 3(10)(1)2 + 13 =

1000 + 300 + 30 +1 = 13312.  233 =

(20 + 3)3 = 203 + 3(20)2(3) + 3(20)(3)2 + 33 = 4000 + 3600 + 180 + 27 = 7807

3.  1013 = (100 + 1)3 = 1003 + 3(100)2(1) + 3(100)(1)2 + 13 =

1000000 + 30000 + 300 + 1 = 1030301

I’ve Got Your Number #6f

Use the cube of the sum pattern to simplify the following expressions. (t + 5)3 =

t3 + 3(t2)(5) + 3(t)(52) + 53 = t3 + 15t2 + 75t + 125

(w + 2) 3 = w3 + 3(w2)(2) + 3(w)(22) + 23 = w3 + 6w2

+ 12w + 8

I’ve Got Your Number #7-9

Pattern 2: Square of a Sum (x + y) 2 = x2 + 2xy + y2

Pattern 4: Square of a Difference (x – y) 2 = x2 - 2xy + y2

Pattern 5: Cube of a Difference (x - y)3 = x3 - 3x2y + 3xy2 - y3

Pattern 6: Conjugates (x + y)(x – y) = x2 – y2

I’ve Got Your Number #7-101. (101)(99) =

(100 + 1)(100 – 1) = 1002 – 12 = 10000 – 1 = 9999

2. (22)(18) = (20 + 2)(20 – 2) = 202 – 22 = 400 – 4 = 396

3. (45)(35) = (40 + 5)(40 – 5) = 402 – 52 = 1600 – 25 = 1575