Download Students will identify and apply real number properties.

SCHOOLS THAT LEAD THE WAY
Kentucky Association of School Councils Conference
September 22, 2009
Williamstown Jr/Sr High School
Contact Us
Ann Rourk - [email protected]
Deana Cummins - [email protected]
Misty Buchanan – [email protected]
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
OR
score 0.5 point
Part b:
score 1 point
OR
score 0.5 point
Part c:
score 1 point
OR
score 0.5 point
Part d:
score 1 point
OR
score 0.5 point
correct numerical value and reasoning explained
using the correct property of real numbers
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
correct numerical value and reasoning explained
using the correct property of real numbers
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
correct numerical value and reasoning explained
using the correct property of real numbers
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
correct numerical value and reasoning explained
using the correct property of real numbers
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
OR
score 1 point
Part c:
score 1 point
graph itself is not labeled, or scale is
incorrect, however, all three equations are
graphed correctly
correct answer with reasoning explained
OR
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.
Cost
Internet Service Provider Plans
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
Plan 1
Plan 2
Plan 3
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
OR
score 0.5 point
Part d:
score 1 point
correct lengths of all 4 sides without perimeter
correct naming of quadrilateral with correct
justification
OR
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.
m AB 
3
3
3
3
; mBC  ; mCD  ; m AD 
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
Substitute into the nth term formula.
an = a1 + (n – 1)d
a1 = _______
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
d = _______
MULTIPLYING POLYNOMIALS
Product of First
Terms
(b + 8)(b + 2)
(x + 9)(x – 4)
(4x + 7)(3x – 8)
(3y + 6)(y – 2)
+
Product of
Outer Terms
+
Product of
Inner Terms
+
=
+
+
+
=
+
+
+
=
+
+
=
+
+
+
=
+
+
+
=
+

=
+
+
+
=
+
+
+
=


=
+
+
+
=
+
+
+
=

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)
Product of
Last Terms
How do I find
the inverse of
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
EXAMPLE
f(x) = 3x2 - 8
What are the
three basic
trigonometry
functions?
B
sin A 
opposite leg
hypotenuse
sin A = _____
sin B = _____
adjacent leg
hypotenuse
cos A 
c
a
cos A = _____
cos B = _____
tan A 
opposite leg
adjacent leg
tan A = _____
A
b
tan B = _____
C
Chief SohCahToa
Graphic Organizer by Dale Graham and Linda Meyer
Thomas County Central High School
Thomasville, Ga.