Schools That Lead the Way

SCHOOLS THAT LEAD THE WAY

Kentucky Association of School Councils Conference September 22, 2009 Williamstown Jr/Sr High School

Graphic Organizers – www.sw-georgia.resa.k12.ga.us/mathk12.html

Page 1 Open-Response Question

The Values of Integers In the table below, p, t, v, and z each represent a different integer. If v  0, find each of the following values. Explain your reasoning using the real number properties.

v  0

v  p = v

t + p = p

v + 5 = t

8  z = p

a) What is the numerical value of p? Write the equation or equations you used to find its numerical value and explain your reasoning using the real number properties.

b) What is the numerical value of t? Write the equation or equations you used to find its numerical value and explain your reasoning using the real number properties.

c) What is the numerical value of v? Write the equation or equations you used to find its numerical value and explain your reasoning using the real number properties.

d) What is the numerical value of z? Write the equation or equations you used to find its numerical value and explain your reasoning using the real number properties. Scoring Guide

SCORE DESCRIPTION

4 Student scores 4 points.

3 Student scores 3 – 3.5 points.

2 Student scores 2 – 2.5 points.

1 Student scores 0.5 – 1.5 points.

0 Student’s response is totally incorrect or irrelevant.

Blank No student response. Score Points

Part a: score 1 point correct numerical value and reasoning explained using the correct property of real numbers OR score 0.5 point correct numerical value and reasoning explained using the incorrect property of real numbers OR correct numerical value and no reasoning is used to explain the value

Part b: score 1 point correct numerical value and reasoning explained using the correct property of real numbers OR score 0.5 point correct numerical value and reasoning explained using the incorrect property of real numbers OR correct numerical value and no reasoning is used to explain the value

Part c: score 1 point correct numerical value and reasoning explained using the correct property of real numbers OR score 0.5 point correct numerical value and reasoning explained using the incorrect property of real numbers OR correct numerical value and no reasoning is used to explain the value Part d: score 1 point correct numerical value and reasoning explained using the correct property of real numbers OR score 0.5 point correct numerical value and reasoning explained using the incorrect property of real numbers OR correct numerical value and no reasoning is used to explain the value Open-Response Question

The Values of Integers In the table below, p, t, v, and z each represent a different integer. If v  0, find each of the following values. Explain your reasoning using the real number properties.

v  0

v  p = v

t + p = p v + 5 = t

8  z = p

a) What is the numerical value of p? Write the equation or equations you used to find its numerical value and explain your reasoning using the real number properties. b) What is the numerical value of t? Write the equation or equations you used to find its numerical value and explain your reasoning using the real number properties. c) What is the numerical value of v? Write the equation or equations you used to find its numerical value and explain your reasoning using the real number properties.

d) What is the numerical value of z? Write the equation or equations you used to find its numerical value and explain your reasoning using the real number properties.

Program of Studies: MA-HS-NPO-S-PNO1 “Students will identify and apply real number properties.”

Program of Studies: MA-HS-NPO-S-PNO4 “Students will justify the solution steps in simplifying expressions or solving an equation.”

Core Content Code: 1.5.1 “Students will identify real number properties (commutative properties of addition and multiplication, associative properties of addition and multiplication, distributive property of multiplication over addition and subtraction, identity properties of addition and multiplication and inverse properties of addition and multiplication) when used to justify a given step in simplifying an expression or solving an equation.” Open-Response Question

Correct Answers

The Values of Integers In the table below, p, t, v, and z each represent a different integer. If v  0, find each of the following values. Explain your reasoning using the real number properties. v  0

v  p = v

t + p = p v + 5 = t

8  z = p

a) What is the numerical value of p? Write the equation or equations you used to find its numerical value and explain your reasoning using the real number properties.

p = 1 v · 1 = v due to the identity of multiplication, a · 1 = a, any number multiplied by 1 equals that number b) What is the numerical value of t? Write the equation or equations you used to find its numerical value and explain your reasoning using the real number properties.

t = 0 0 + p = p due to the additive identity property, a + 0 = a, any number added to 0 equals that number c) What is the numerical value of v? Write the equation or equations you used to find its numerical value and explain your reasoning using the real number properties.

v = 5 Since you know t = 0, the equation v + 5 = 0 can be used to solve for v. If v + 5 = 0, then v must equal  5 to have the equation equal 0, due to the additive inverse property. The sum of any number and its opposite is zero. d) What is the numerical value of z? Write the equation or equations you used to find its numerical value and explain your reasoning using the real number properties.

z = Since you know p = 1, the equation 8  z = 1 can be used to solve for z. If 8  z = 1,

then z must equal to have the equation equal 1, due to the multiplicative inverse property. Any number multiplied by its inverse equals 1.

Open-Response Question

Internet Service Provider

Megan wants to change her Internet Service Provider. She is considering three different plans: Plan 1 charges a $15 monthly fee, plus $0.08 per minute of use.

Plan 2 charges a $5 monthly fee, plus $0.11 per minute of use.

Plan 3 charges a flat monthly fee of $49.95. a) For each plan, write an algebraic equation that represents the monthly

cost, C, for m minutes per month.

b) Graph each of the three equations on the same coordinate axes (with correct labels and scales). Label each line.

c) Megan expects to use 500 minutes per month. In which plan do you think Megan should enroll? Explain your reasoning.

Scoring Guide

SCORE DESCRIPTION

4 Student scores 4 points. 3 Student scores 3 – 3.5 points.

2 Student scores 2 – 2.5 points.

1 Student scores 0.5 – 1.5 points.

0 Student’s response is totally incorrect or irrelevant.

Blank No student response. Score Points

Part a: score 1.5 points .5 points each for correct equation given for each plan

Part b: score 1.5 point .5 points each for correct graph of each plan

score 1 point graph itself is not labeled, or scale is incorrect, however, all three equations are graphed correctly

Part c: score 1 point correct answer with reasoning explained

score 0.5 point correct answer with no explanation

provided

Open-Response Question

Internet Service Provider

Megan wants to change her Internet Service Provider. She is considering three different plans:

Plan 1 charges a $15 monthly fee, plus $0.08 per minute of use. Plan 2 charges a $5 monthly fee, plus $0.11 per minute of use.

Plan 3 charges a flat monthly fee of $49.95. a) For each plan, write an algebraic equation that represents the monthly

cost, C, for m minutes per month.

b) Graph each of the three equations on the same coordinate axes (with correct labels and scales). Label each line.

c) Megan expects to use 500 minutes per month. In which plan do you think Megan should enroll? Explain your reasoning.

Core Content Code: 5.3.3 “Students will model, solve and graph first degree, two-variable equations and inequalities in real-world and mathematical problems.”

Program of Studies: MA-HS-AT-S-EI17 “Students will write and solve linear sentences, describing real-world situations by using and relating formulas, tables, graphs and equations.”

Open-Response Question Correct Answers

Internet Service Provider

Megan wants to change her Internet Service Provider. She is considering three different plans:

Plan 1 charges a $15 monthly fee, plus $0.08 per minute of use. Plan 2 charges a $5 monthly fee, plus $0.11 per minute of use. Plan 3 charges a flat monthly fee of $49.95. a) For each plan, write an algebraic equation that represents the monthly cost, C, for m minutes per month.

Plan 1: C = 15 + 0.08m Plan 2: C = 5 + 0.11m Plan 3: C = 49.95

b) Graph each of the three equations on the same coordinate axes (with correct labels and scales). Label each line.

Internet Service Provider Plans 70 65 60 55 50 45

t 40 s 35 o Plan 1

C 30 25 Plan 2 20 Plan 3 15 10 5 0 0 50 100 150 200 250 300 350 400 450 500 550

Minutes

c) Megan expects to use 500 minutes per month. In which plan do you think Megan should enroll? Explain your reasoning.

Megan should enroll in Plan 3. The graph in part b shows that at 500 minutes, she would be paying $49.95 for plan 3, about $60 for Plan 2, and about $55 for Plan1. Therefore, plan 3 is the most economical plan.

Open-Response Question

Quadrilateral

Consider the following points: A(-2,1) , B( 2, - 2) , C( 8,7) , D ( 4,10)

a) Graph the quadrilateral ABCD. b) Find the slope of each side of the quadrilateral.

c) Find the perimeter of quadrilateral ABCD.

d) What is the most precise name for the quadrilateral ABCD? Justify your choice.

Scoring Guide

SCORE DESCRIPTION

4 Student scores 4 points.

3 Student scores 3 – 3.5 points.

2 Student scores 2 – 2.5 points. 1 Student scores 0.5 – 1.5 points.

0 Student’s response is totally incorrect or irrelevant.

Blank No student response.

Score Points

Part a: score 1 point correct graphing of all 4 ordered pairs

Part b: score 1 point correct slope of all 4 sides

Part c: score 1 point correct lengths of all 4 sides and correct perimeter

score 0.5 point correct lengths of all 4 sides without perimeter

Part d: score 1 point correct naming of quadrilateral with correct justification

score 0.5 point only correct naming of quadrilateral Open-Response Question

Quadrilateral

Consider the following points: A(-2,1) , B( 2, - 2) , C( 8,7) , D ( 4,10)

a) Graph the quadrilateral ABCD.

b) Find the slope of each side of the quadrilateral. c) Find the perimeter of quadrilateral ABCD.

d) What is the most precise name for the quadrilateral ABCD? Justify your choice.

Program of Studies: MA-HS-G-S-CG1 “Students will use the coordinates of two points on a line to determine its slope and use slope to express the parallelism and perpendicularity of lines.”

Program of Studies: MA-HS-G-S-CG3 “Students will find the distance between two points using their coordinates and the Pythagorean theorem or the distance formula.”

Program of Studies: MA-HS-G-S-CG6 “Students will use Cartesian coordinates and other coordinate systems (e.g., navigational, polar, spherical systems) to analyze geometric situations.”

Core Content Code: MA-HS-3.3.1 “Students will apply algebraic concepts and graphing in the coordinate plane to analyze and solve problems (e.g., parallel and perpendicular lines, the distance between two points, the slope of a segment).” Open-Response Question

Correct Answers

Quadrilateral Consider the following points: A(-2,1) , B( 2, - 2) , C( 8,7) , D ( 4,10)

a) Graph the quadrilateral ABCD. b) Find the slope of each side of the quadrilateral. -3 3 -3 3 mAB = ; mBC = ; mCD = ; mAD = 4 2 4 2 c) Find the perimeter of quadrilateral ABCD. AB = 5 ; BC = 117 ; CD = 5 ; AD = 117

P =2(5) + 2( 117) = 10 + 2 117 = 31.63 d) What is the most precise name for the quadrilateral ABCD? Justify your choice. The most precise name for quadrilateral ABCD is PARALLELOGRAM. Both pair of opposite sides are parallel, but slopes are not opposite reciprocals, so the angles are NOT right angles. Therefore, the quadrilateral can’t be rectangle. In addition, the lengths of the four sides are not congruent, so it is not a rhombus. Writing an Equation for an Arithmetic Sequence

a1 = first term d= common difference th n = n term of the sequence an = a1 + (n – 1)d

an = a1 + (n – 1)d Formula for the nth term of an Arithmetic Sequence

Follow the steps below to write an equation to represent the arithmetic sequence described.

Loans: The following arithmetic sequence represents the amount of money that Ed owes his brother each week after he borrows the same amount:

2, -2, -6, -10, …

a) Write an equation for the nth term of the sequence.

Determine the common difference

th Substitute into the n term formula. a1 = ______d = ______an = a1 + (n – 1)d

Simplify the equation for the nth term.

b) Use the equation to find how much Ed owes his brother after 25 weeks.

Replace n with 25 in your equation for the nth term. Simplify.

Graphic Organizer by Ann Rourk Williamstown Jr/Sr High School

MULTIPLYING POLYNOMIALS Product of First Product of Product of Product of Last + + + Terms Outer Terms Inner Terms Terms

= + + +

(b + 8)(b + 2) = + + +

= + +

= + + +

(x + 9)(x – 4) = + + +

= + 

= + + +

(4x + 7)(3x – 8) = + + +

=  

= + + +

(3y + 6)(y – 2) = + + +

= 

Try the following on your own:

1) (x + 2)(x + 3) 2) (p – 4)(p + 2)

3) (3x + 1)(4x + 3) 4) (a – 3b)(2a 5b) How do I find EXAMPLE the inverse of f(x) = 3x2 - 8 a function?

Switch to the y = notation from the f(x) =.

Exchange x and y in the problem and solve for y.

Rewrite as f-1(x) = .

Graphic Organizer by Dale Graham and Linda Meyer Thomas County Central High School; Thomasville GA

B opposite leg What are the sin A  hypotenuse three basic trigonometry sin A = _____ functions? sin B = _____

adjacent leg cos A  hypotenuse c cos A = _____ a cos B = _____

opposite leg tan A  adjacent leg

tan A = _____

tan B = _____ A b C Chief SohCahToa

Graphic Organizer by Dale Graham and Linda Meyer Thomas County Central High School Thomasville, Ga.