Algebra Linguists
DVC Mathematics Conference
February 28, 2014
Michelle Neibch, Michelle.Neibch@cusd200.org
Angela Raso, Angela.Raso@cusd200.org
Wheaton North High School
Wheaton, IL
This handout will be available at http://www.cusd200.org/Page/9152
Table of Contents
1. Matching Activities and associated homework assignments *
Quadratics, Lines, Functions
2. Which one of these is not like the others
Polynomials and Quadratics
3. Notecard Activity -- Give half the class a card with a quadratic function in standard form, the
other half a card with a quadratic function in intercept form, students find their partner, then
work together to come up with the vertex form and make a graph.
4. Finding Point P and Treasure Hunt
Algebra 2 semester review, Algebra 1 introduction to systems
5. Activities about Solving Systems
Systems clue problems, Non-linear systems
6. Algebra/Geometry Connections, Generalizing
Squares in a Rectangle, Area and Transformations of Functions, Exponents
7. Brainstorming Ideas
* We want to make sure to thank some members of our department for contributing to these materials.
Specifically, J.T. Johns created the Lines matching activity and Amy Pekkarinen created the Functions
matching activity.
Name:_______________________Date:_________
Quadratic Function Matching Activity
Match your quadratics cards appropriately, and complete the following table.
Description Standard
of function
Form
A
D
N
G
Q
F
Vertex
Form
Factored
Form
Graph
Name:_______________________Date:_________
Quadratic Function Matching Activity
Match your quadratics cards appropriately, and complete the following table.
Description Standard
of function
Form
Vertex
Form
Factored
Form
Graph
A
7
j
θ
∞
D
2
t
β
√
N
8
b
π
+
G
3
r
ε
$
Q
5
m
Δ
#
F
6
h
λ
!
Line Experts: _______________________________________________________________________
Lines matching activity
Use the cards provided to complete the chart. Fill in the number for each category.
Slope-intercept
form (SI)
1
2
3
4
5
6
7
8
9
10
11
12
Standard form
(Stan)
Point-slope form
(P-S)
Graph of line
(G)
Description of
Line (D)
(SI) 1
y  2x  3
(SI) 4
(SI) 2
1
y  x 3
2
(SI) 3
y  2x  3
(SI) 5
(SI) 6
y  x
yx
(SI) 7
(SI) 8
(SI) 9
y  x 1
y  x 1
y4
(SI) 10
(SI) 11
(SI) 12
2
y  x 5
3
3
y  x 5
2
1
y  x3
2
x  3
(Stan) 3
(Stan) 12
(Stan) 6
2 x  y  3 x  2 y  6 2 x  y  3
(Stan) 9
(Stan) 10
x  2 y  6 x  y  0
(Stan) 1
x y 0
(Stan) 2
(Stan) 5
(Stan) 11
x  y  1
x  y 1
0x  y  4
(Stan) 7
(Stan) 8
(Stan) 4
x  0 y  3 2 x  3 y  15 3 x  2 y  10
(P-S) 4
y  1  2( x  2)
(P-S) 9
(P-S) 11
1
y  1   x  4
2
(P-S) 8
y  5  2( x  1)
(P-S) 5
(P-S) 1
y  0  1( x  0)
y  0  1( x  0)
(P-S) 10
(P-S) 6
(P-S) 7
y  6  1( x  5)
y  4  1( x  5)
y  4  0( x  2)
(P-S) 3
(P-S) 12
(P-S) 2
1
y  5   x  4
2
0  1( x  3)
y 1 
2
 x  6
3
3
y  4  ( x  6)
2
(G) 2
(G) 11
(G) 1
(G) 7
(G) 12
(G) 8
(G) 9
(G) 6
(G) 3
(G) 5
(G) 10
(G) 4
(D) 9
(D) 2
(D) 8
This is a line with a slope
of two that passes through
the point (2,1).
This is a line with a slope
of one-half and a yintercept at (0,-3).
The output is three more
than twice the input.
(D) 5
(D) 12
(D) 4
This is a line with a slope
of one-half that passes
through the point (4,5).
This is a line through the
origin with a slope of
negative one.
The output is the same
value as the input.
(D) 1
(D) 10
(D) 11
The output is one more
than the input.
This is a line with a slope
of one and a y-intercept at
(0,-1).
This is a horizontal line.
(D) 6
(D) 3
(D) 7
This is a vertical line.
This line has a slope of
two-thirds.
This is a line with a
positive slope that passes
through the point (6,4).
Mathlete:___________________________________________________________Date_____________
Complete the missing sections of the table below.
Name a Point
Standard Form
Slope Intercept
other than yForm
intercept on the
line and the
slope
2 x  6 y  6
Point:
not y-int!
Slope:
Point: (1, 5)
Slope: 2
y
1
x4
2
Point:
Slope:
Point:
Slope:
Point-Slope Form
Graph
Mathematicians:__________________________________________________Date:______________
Function Matching
Match each verbal description of a function with the coordinating table, equation, and graph. Write the letter or
symbol in the chart below.
Verbal Description
A.
E.
Q.
O.
.
U.
G.
@.
Table
Equation
Graph
Mathematicians:__________________________________________________Date:______________
Function Matching
Match each verbal description with the coordinating table, equation, and graph. Write the letter or symbol in
the chart below.
Verbal Description
Table
Equation
Graph
A.
P.
Z.
S.
E.
$
L.
V.
Q.
K.
X.
H.
O.
T.
J.
Y.
.
W.
%
I.
U.
#
F.
C.
G.
D.
R.
M.
@.
B.
N.
!
y
x
P.
2
1
A. The output is the
input tripled less one.
0
1
2
f ( x)
7
4
1
Z.
f ( x)  3x  1
S.

x


2
5

y
x
$.
2
E. The output is the
reciprocal of the
input.
K.
Q. The output is four
more than the
absolute value of the
input.
.
The input is the
square of the output.
#
U. The output is one
more than the square
of the input.
1
2
1
1
0
1
2
undef.
1
2
1
0
1
2
6
5
4
5
6
x
f ( x)
2
1
8
6
4
2
0
x
f ( x)
0
1
4
9
16
0
1
2
3
4
x
f ( x)
2
1
5
2
1
2
5
0
1
2
D.
G. The output is the
sum of the input and
one squared.
L.
1
f ( x) 
x

V.
x


1
2
f ( x)
0
1
2
W.

x
T.
O. The output
decreases by 2 when
the input increases by
one. f (0)  4
f ( x)

y
X.
H.
f ( x) | x |  4

x


y

J.
f ( x)  2 x  4

Y.
x


y

%
f ( x)  x
I.

x


y

F.
f ( x)  x 2  1

C.
x



y
x
f ( x)
2
1
1
0
1
4
9
0
1
2
R.
f ( x)   x  1
M.

2
x



y
B.
@ The output is the
opposite of the input.
x
f ( x)
2
1
2
1
0
0
1
2
1
2
N.
f ( x)   x

!
x



One of These Things is Not Like the Others
For each set, find the 1 that is NOT like the others. In the polynomial sets, all but one are equivalent forms of the same polynomial
expression. It’s your job to find the one that’s not equivalent. Circle the one that’s different. Some sets are not polynomials.
1.
2.
2( x  5)( x  5)
2 x 2  50
(4 x 2  30)  (2 x 2  20)
(3x 2  6 x  32)  (1x 2  6 x  82)
2(25  x 2 )
3.
4.
x 12 x  32
6( x  1)  x( x  1)
( x  6)2  5
(2 x 2  6 x  4)  (11x  x 2  10)
1(32  12 x  x 2 )
( x  6)( x  1)
x 2  4 x  8 x  32
2 x 2  10 x  12
(5x 2  4 x  22)  (4 x 2  8x  10)
x x 5 x 2 3
5.
6.
2
(3  x)(4)(2 x  5)
( x  4)2  36
4(3  x)(5  2 x)
( x  4  6)( x  4  6)
4( x  3)(2 x  5)
x 2  8 x  20
8 x 2  4 x  60
( x  2)( x  10)
(5  2 x)( x  3)(4)
x( x  2)  10( x  2)
7.
8.
3( x  4)(2 x  1)
(5 x3  4  2 x  8 x 2 )  (4 x 2  8 x  7 x3  2 x3 )
6 x3  21x  12
4 x2  6 x  4
3(2 x 2  7 x)  3(4)
2(2 x 2  3x  2)
( x  4)(6 x  3)
4( x  1)( x  1)  6 x
(2 x 2  7 x  4)3
4( x 2  1)  6 x
9.
10.
( x  3)  6( x  3)
2
2
7( x  3) 2
7( x 2  9)
(3  x)(3  x)  ( x 2  6 x  9)6
7 x 2  42 x  63
11.
(3x  15)(2 x  10)
( x  2( x  5))( x  3( x  5))
(3x  10)(2 x  15)
1(10  3x)(15  2 x)
6 x 2  65 x  150
12.
16
5
1
3
5
3.2
320%
9
5
13.
(3 x  1)(2 x  4)
14.
2 x( x 2  5 x  1)  4(2 x 2  6 x  4)
6 x 2  14 x  4
2 x3  18 x 2  26 x  16
6 x 2  2 x  12 x  4
2 x3  10 x 2  2 x  8 x 2  24 x  16
2 x(3x  1)  4(3 x  1)
2 x 2 ( x  9)  2(13x  8)
(3 x  1)( x  2)
2 x( x 2  13)  2(9 x 2  8)
Mathematician:_____________________________________________Date:_____________Binder:
One of These Things is Not Like the Others-Quadratic Edition
For each set, find the 1 that is NOT like the others. In most of the sets below, all but one are equivalent forms of the same quadratic
function. It’s your job to find the one that’s not equivalent. Circle the one that’s different. Be sure to be able to justify your choice.
Some sets include graphical or verbal representations of functions and some sets don’t involve quadratics at all!
1.
2.
y
1 2
x 5
2
y
1
( x  0) 2  5
2
y5 
3.
1 2
x
2
y
1
( x)( x)  5
2
y
1 2
x  5x  5  5x
2
4.
y  x  10 x  20
y  x 2  4 x  12
y  x 2  10 x  25  20  25
y  ( x  2)( x  6)
y  25  x 2  10 x  25  20
y  ( x  2)( x  6)
y  ( x  5)2  5
y  x 2  4 x  4  12  4
y  ( x  5)2  5
y  ( x  2)2  16
5.
6.
2
y  2( x  3)( x  2)
y  x 2  16
y  (2 x  6)(2 x  4)
y  ( x  4)2
y  2 x 2  2 x  12
y  ( x  4)( x  4)
y  2( x 2  x  6)
y  x 2  8 x  16
2
1  25

y  2  x   
2
2

y  16  x 2  8 x
Go back and make a rough sketch the parabolas represented in #’s 2 – 6!
Label the vertex of each parabola in each sketch.
7.
y
y
y  2( x  5)( x  5)
x


x

y  2 x 2  50
y  2( x  0)2  50



8.
y
y  2( x  3)2  4

y  2( x 2  6 x  7)
x



y  2 x 2  12 x  14
9.
10.
2
y  x4
5
2 x  5 y  20
2
y  2  ( x  5)
5
5 y  2 x  20
y4
11.
y
y  2( x 2  6 x  9)  14  18
25
100
2.5
5
10
250%
2
2
x
5
10
4
12.
y  ( x  3)( x  5)
1 2
x 8
2
A parabola with x-intercepts of
(3, 0) and (5, 0).
1
y  ( x  4)( x  4)
2
A parabola with axis of symmetry of x = 1.
y 8 
1 2
x
2
A parabola with a vertex of (1, -16).
2 y  x2  16
y

y
x



x






Finding Point P…
Point P is the intersection of lines AB and CD. Points A, B, C and D can be obtained by completing the
“clues,” or problems. Your work should all be done without a calculator and should include clear steps so I can
see your work and so that you can use this as a review for your exam. Remember, all of these topics are on the
final exam so keep in mind that you and your partner/group need to understand how to do each section. Have
fun!!
Section 1:
Point A is the vertex of the parabola formed by the quadratic function y = 2x2 – 12x + 19
To find Point B you will need to find the inverse of the function, f(x) = (1/7)x + (5/7) . The inverse function
will be a linear function where the slope of the inverse function is the x-coordinate of Point B and the yinterceptof the inverse function is the y-coordinate of Point B.
Point A: __________________
Point B: _____________________
Section 2:
Point C is the smallest x-intercept of the polynomial function, f(x) = x4 – 26x2 + 25.
Given that the point (3, -10) is on f(x), where does that point lie on y = ½ f(-x + 13) + 2 ? This
is Point D.
Point C: ___________________
Point D: ____________________
Section 3:
Find the equation of Line AB:
_______________________________
Find the equation of Line CD:
_______________________________
Section 4:
Find Point P! Point P is ______________________
Sections Reviewed in this Activity:
1. Linear Functions (writing the equations of lines AB and CD, and knowing slope and yintercept of a linear function for Point B)
2. Inverse Functions (finding the inverse of the function for Point B)
3. Quadratic Functions (finding the vertex of a parabola, solving quadratic equations after
factoring for Point C)
4. Transformations of Functions (transforming the point (3, -10) for Point D)
5. Solving Systems of Equations (finding Point P, which is the solution to the system of
equations of Lines AB and CD)
6. Polynomial Functions (knowing how to find the x-intercept of a polynomial function)
Treasure Hunter: _______________________________
Algebra B Treasure Hunt
Captain Jack is looking for buried treasure on a small island in the South Pacific.
He has a treasure map with clues drawn by pirates hundreds of years ago.
You must correctly locate each point on the treasure map before you will be
given the next clue to help Captain Jack.
Let the search begin!
According to the map, Captain Jack is supposed to start at the rock shaped like a
dragon, which on the map is labeled point D.
The x-value of point D can be found by solving the equation:
1
x3 7x
4
The y-value of point D is thirteen less than twice the x-value found above.
The coordinates of point D are: (
,
)
The treasure map says to next go to the ship wrecked boat,
which on the map is called point S
The x-value of point S is the slope of the line through points:
(-5, -6) and (-8, 15)
The y-value of point S is the slope that is parallel to the slope found above.
The coordinates of point S are: (
,
)
,
)
Next you must find point T, where the trees form the Triangle of Truth.
Point T is the x-intercept of the equation:
1
y  4   (x  18)
3
The coordinates of point T are: (
The treasure map says the last point to find before locating the treasure is Point V.
This is the mysterious vanishing point of the island.
Solve the absolute value equation: 2|2 p  3|8  18
The x-value of point V is the lesser of the two answers for p.
The y-value of point V is the greater of the two answers.
The coordinates of point V are: (
,
)
Now that you have located all four of the island locations, you need to come up with
equations of lines between them.
Find the equation of line DS
Equation of line DS:
Find the equation of line TV
Equation of line TV:
Graph line DS and line TV on the treasure map below. Label their intersection point X
X marks the spot! Congratulations on locating the treasure 
The coordinates of the treasure (point X) are: (
,
)
Name:______________________________________________Date:__________________
SYSTEMS CLUE PROBLEMS TAKE 2
Set up and SOLVE the system using the information provided. Write out all work neatly.
Line 1: Contains the points (4,– 3 ) and (6,– 3 ).
Line 2: Is perpendicular to 4x – 2y = 8 and contains (– 3, 4)
Work for Line 1:
Write Line1’s equation here:
It can be left in any form you choose.
Work for Line 2:
Write Line 2’s equation here:
It can be left in any form you choose.
Solve the system
Write the solution as an ordered pair.
Name:______________________________________________Date:__________________
SYSTEMS CLUE PROBLEMS TAKE 3
Set up and SOLVE the system using the information provided.
Line 1: Contains the points (4, 3) and (2, 5).
Line 2: Has an x intercept of – 3 and a y intercept of – 3
SYSTEMS CLUE PROBLEMS TAKE 4
Set up and SOLVE the system using the information provided.
Line 1: Is perpendicular to 4x – 5y = 1 and has an x intercept of – 4
Line 2: Contains (-8, 3) and (-8, 11)
Non-Linear Systems of Equations
Help your students see beyond the standard linear systems!
Directions: Solve the following systems of non-linear equations, and list
all solutions as ordered pairs.
#1)
x  y2  4

 2
2

 x  y  16
#2)
1
2

 x 2 y 2  11


 4  2  14
 x 2 y 2
Now going back to linear systems, have your students solve a
system of linear equations with all variables as coefficients. This
makes them think about the algebra behind what they are doing
when they use substitution or elimination.
Directions: Solve for x and y in terms of a, b, c, d, e and f.
#3)
ax  by  c

dx  ey  f
Squares in a Rectangle
The rectangle below has been divided into eleven squares. Write an equation below that will yield the value of x,
then solve for x. Start by labeling all of the sides of the squares next to each bracket in terms of x.
Equation:______________________________________________________________________
Solution:_x = ___________ Check your answer by plugging it in to some of your other expressions to make
sure that your answer makes sense for the figure above.
Given… f(x) = x
The task: Determine the area formed between the graph of the function and the x-axis over the
interval [1, 3] for the following.
#1)
a.
b.
c.
d.
e.
f.
f(x)
f(2x)
f( ½ x)
f(3x)
f(4x)
f(bx) ; b > 0
#2)
a.
b.
c.
d.
e.
f.
f(x)
f(x) + 1
f(x) + 2
f(x) +3
f(x) + 4
f(x) + d ; d > 0
#3)
a.
b.
c.
d.
e.
f.
f(x)
f(x + 1)
f(x + 2)
f(x + 3)
f(x + 10)
f(x + c) ; c > 0
#4)
a.
b.
c.
d.
e.
f.
f(x)
2f(x)
3f(x)
10f(x)
25f(x)
a•f(x) ; b > 0
#5)
a • f(b(x + c)) + d
Algebra 2A
Mathematician________________________________
*Simplify each problem completely.
#1)
2n 4  2(2n )
2(2n3 )
4
#2)
Algebra 2A
 3 6 a9   6 3 a9 

 


 

4
1
#3)  2  3(2  3) 
1
Mathematician________________________________
*Simplify each problem completely.
#1)
2n 4  2(2n )
2(2n3 )
4
#2)
 3 6 a9   6 3 a9 

 


 

4
1
#3)  2  3(2  3) 
1
Brainstorming:
Use this space to brainstorm ways that you could use the ideas
presented today in your own classroom.