2012-2013 Enhanced Instructional Transition Guide Mathematics Algebra I Unit 09

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Nov 29, 2013 (3 years and 8 months ago)

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Algebra
1/Mathematics

Unit 0
9
: Possible Lesson 0
1

Suggested Duration: 9

days


©2012
, TESCCC

04/21
/13

page
1

of
9

2012
-
2013 Enhanced Instructional Transition Guide

Mathematics Algebra I Unit 09

Unit 0
9
:

Quadratic Functions

(9

days)


Possible Lesson 01

(
9

days)



POSSIBLE LESSON 01 (
9

days)


Lesson Synopsis:


Students
analyze the
characteristics and graphs of quadratic functions. Students interpret and describe the effects of changes in the parameters o
f quadratic
functions.

Students investigate real
-
world situations represented by quadratic functions.


TEKS:

A.1

Foundations for
functions. The student understands that a function represents a dependence of one quantity on another and can be described in

a variety of ways.
The student is expected to:


A.1D

Represent relationships among quantities using concrete models, tables, grap
hs, diagrams, verbal descriptions, equations
, and inequalities
.
Readiness Standard

A.1E

Interpret and make decisions, predictions, and critical judgments from functional relationships.
Readiness Standard


A.2

Linear functions. The student understands the

meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and
describes the effects of changes in parameters of linear functions in real
-
world and mathematical situations. The student is expected to
:


A.2A

Identify and sketch the general forms of

linear (
y
=
x
) and

quadratic (
y
=
x
²) parent functions.

Supporting Standard

A.2B

Identify mathematical domains and ranges and determine reasonable domain and range values for given situations, both continuo
us and discrete.
Readiness
Standard

A.2C

Interpret situations in terms of given graphs or create
s

situations that fit given graphs.
Supporting Standard

A.2D

Collect and organize data, make and interpret scatterplots
(including recognizing positive, negative, or no correlation for data approximating linear situations)
, and
model, predict, and make decisions and critical judgments in problem situations.
Readiness Standard


A.9

Quadratic and other nonlinear functions. The student understands that the graphs of quadratic functions are affected by the p
arameters of the fu
nction and can
interpret and describe the effects of changes in the parameters of quadratic functions. The student is expected to:


A.9A

Determine the domain and range for quadratic functions in given situations.
Supporting Standard

A.9B

Investigate, describe, and predict the effects of changes in
a

on the graph of
y
=
ax
2

+ c
.
Supporting Standard

A.9C

Investigate, describe, and predict the effects of changes in
c

on the graph of
y
=
ax
2

+ c
.
Supporting Standard

A.9D

Analyze graphs of quadratic functions and draw conclusions.
Readiness Standard


Performance Indic
a
tor(s):

Create a mini
-
notebook that explains the effects of changes in “
a
” and “
c
” on the graph of
2
y ax c
 
. The notebook should include
the following:



Graphical representations



Symbolic representations



Verbal descriptions



Transformations


Reflections (across the
x
-
axis), translations (up and down only), and dilations (stretches and compressions)

(A.2A; A.9A, A.9B, A.9C, A.9D)


1E; 3H; 5F


Analyze a problem situation that can be represented by a quadratic function such as the following:




Pablo wants to build a pen for the pig he is raising in his agriculture class. Pablo only has 36 feet of fencing. What dimens
ions can Pablo use

to give the pen
the maximum area?


Algebra
1/Mathematics

Unit 0
9
: Possible Lesson 0
1

Suggested Duration: 9

days


©2012
, TESCCC

04/21
/13

page
2

of
9

2012
-
2013 Enhanced Instructional Transition Guide

Mathematics Algebra I Unit 09


Create a graphic organizer that includes a table, scatterplot, parent function, and representative function. Use the represen
tations to determine characteristics of
the function, including independent/dependent variables, domain/range, continuous/discrete
data, increasing/decreasing intervals, vertex, minimum/maximum, axis
of symmetry,
y
-
intercept, and
x
-
intercepts. Explain the meaning of each characteristic in terms of the problem situation. Justify the pen Pablo should build
in order
to yield the maximum
area. (A.1D, A,1E; A.2A, A.2B, A.2C, A.2D; A.9A, A.9D)


1C; 3H


Key Understanding(s):



Changing the parameter values,
a

and
c,

in the function

y

=
ax
2

+
c
, transforms the graph of the quadratic parent function. The parameter value
a

defines a
reflection and/or dilation; whereas in contrast, the
c

defines the translation, up or down.



Quadratic functions have specific characteristics that define the function and can be represented using various methods.



Some data collection situations
are modeled by quadratic functions, and various representations of the function can be used to describe, predict, and justify

values for the problem situation over an appropriate domain and range.



Misconception(s):



Some students may think that when eval
uating
2
( )
f x x
 
, the negative is also square
d

when actually only the substituted
x

value is squared.


Vocabulary of Instruction:



area



continuous



dilate



discrete



domain



finite difference



maximum




minimum



parabola



parameter change



parent
function



perimeter



quadratic function




range



reflect



vertex



vertical compression (wider)



vertical shift



vertical stretch (narrower)



Suggested

Day

Suggested
Instructional
Procedure
s

Notes for Teacher

1


Topics:



Modeling the quadratic function


Engage

1

Students
investigate

data that can be modeled by a
quadratic function.

Students analyze the data and its
characteristics using various representations. Students

determine the meaning of the characteristics in the
problem situation.


Instructional
Procedures:

1.

Place students in
pairs.
Distribute handout:
Dive Straight In

to each student.

Display teacher resource:
ATTACHMENTS



Teacher Resource
:
Dive
Straight In

KEY

(
1 per
teacher
)



Teacher Resource
:
Dive
Straight In

(
1 per teacher
)



Handout:
Dive Straight In

(
1
per student
)


MATERIALS



graphing

calculator (1 per

Algebra
1/Mathematics

Unit 0
9
: Possible Lesson 0
1

Suggested Duration: 9

days


©2012
, TESCCC

04/21
/13

page
3

of
9

2012
-
2013 Enhanced Instructional Transition Guide

Mathematics Algebra I Unit 09

Suggested

Day

Suggested
Instructional
Procedure
s

Notes for Teacher

Dive Straight In
,

and facilitate a class discussion of the problem situation.

Ask:



What happens when someone dives off a diving board?

Answers
may

vary
.

Diver goes up higher
than the diving board. The diver then goes down and hits the surface of the water. The diver then
goes under the water for a distance and then comes back up to the surface of the water; etc.



If you were to graph the d
iver’s distance above the water over time, what would the graph look
like?

Answers
may
vary. Have students model the path with their hand
, etc
.



How is the graph the same or different from the ones you have already created?
Answers
may

vary.

The graph is cu
rved; etc.

2.

Instruct

students

to

work

with their partner to complete problems 1


2 on handout:
Dive Straight In
.

Allow students
time to complete the questions. Monitor and assess students
to check for understanding.
Using teacher resource:
Dive Straight In
,

facilitate a class d
iscuss
ion of student results, clarifying any
misconceptions.

Ask:



Do all tables contain the same data values?

(
Yes, because all students used the same data.
)



Are all graphs exactly the same?

(
Yes, because all students used the same da
ta.
)



What is the shape of the graph?

(
p
arabolic
)



What type of function has a parabolic shape?

(
q
uadratic
)



What power is on the leading
x

term that creates the curve of the quadratic function?

(
s
quare
)

3.

Instruct students to work with their partner

to complete questions 3



2
0 on handout:
Dive Straight In
.

Allow students time to complete the questions. Monitor and assess students to check for understanding.
Using teacher resource:
Dive Straight In
, facilitate a class discussion of student results, c
larifying any
misconceptions. E
mphasize the questions pertaining to
x
-

and
y
-
intercepts.


student)



graphing calculator with
display (1 per teacher)


2



3

Topics:



Quadratic parent function



Modeling the quadratic function


Explore/Explain 1

Students

identify the quadratic parent function and its characteristics. Students

explore problem situations
involving dilations and area that can be modeled by quadratic functions.


Instructional Procedures:

Day 2

1.

Place students in pairs. D
istribute handout:
Look

at My Parents

to each student. Distribute 1 sheet of
chart paper, 1 set of chart markers, and 9 sticky dots to each pair of students. Instruct students to work
with their partner to complete the activity
,

and create and post their display. Facilitate a cl
ass discussion
of student posters, clarifying any misconceptions.

ATTACHMENTS



Teacher Resource
:
Look at
My Parents

KEY

(
1 per
teacher
)



Handout:
Look at My Parents

(
1 per student
)



Teacher Resource
:
Dilation
Dilemma

KEY

(
1 per teacher
)



Teacher Resource
:
Dilation
Dilemma

(
1 per teacher
)



Handout:
Dilation
Dilemma

(
1
per student
)



Teacher Resource
:
Working
with Quadratics

KEY

(
1 per
teacher
)


Algebra
1/Mathematics

Unit 0
9
: Possible Lesson 0
1

Suggested Duration: 9

days


©2012
, TESCCC

04/21
/13

page
4

of
9

2012
-
2013 Enhanced Instructional Transition Guide

Mathematics Algebra I Unit 09

Suggested

Day

Suggested
Instructional
Procedure
s

Notes for Teacher

2.

Distribute handout:
Dilation Dilemma
to each student.
Instruct students to work with their partner to
complete problems 1


6.
Allow students time to complete the questions. Monitor and assess students to
check for understanding.
Display teacher resource:
Dilation Dilemma
, and facilitate a class discussion of
student results, clarifying any misconceptions.


3.

Using teacher resource:

Dilation Dilemma
, facilitate a class discussion

on the comparison of
finite
differences

in

linear

and quadratic functions
, modeling problem 7.

Ask:



What kind of graph is this?

(
q
uadratic
)



What is the vertex? What does it represent?
(
f
or this situation,
(0,0); the lowest point of the graph
)



Why would negative numbers not be reasonable

range values

in this situation?
Answers may
vary.
Because the leading coefficient of x
2

is positive and when you square a n
umber, the solution is
positive; etc.



How are the exponents different in linear versus quadratic functions?
(
In linear functions the
exponents are always 1. In quadratic functions, exponents are always 2.
)



Do you think quadratics can be transformed like linear functions
?

Answer may vary.


This

lesson help
s

students understand that the functions transform the same

way
, only the fam
ily
function differ
s
.



Will the same pattern for these parameter changes apply?
Answer may vary.


This lesson help
s

students understand that the parameter changes will be the same, only the family
function differ
s
.

4.

Instruct students to work with their partner to complete problems 8


16 on handout:
Dilation Dilemma
.
This

may
be
complete
d
as homework, if necessary.


Day

3

5.

Facilitate a class discussion to debrief

handout:

Dilation Dilemma
to check for understanding
.

6.

Distribute handout:
Working with Quadratics

to each student.

Instruct students to work independently to
complete the handout. This may be completed as homewor
k, if necessary.




Handout:
Working with
Quadratics

(
1 per student
)


MATERIALS



graphing

calculator (1 per
student)



graphing calculator with
display (1 per teacher)



chart paper (1 sheet per
2
students)



chart markers (1 set per 2
students)



sticky dots (
9 per 2 students)



pattern blocks (green
equilaterals) (10 per 2
students)



color tiles (10 per 2 students)


4

Topics:



Quadratic

functions



Parameter changes



Explore/Explain
2

Students
explore
effects of
parameter changes on the quadratic parent function.



Instructional Procedures:

1.

Distribute handout:
Investigating
y

=
ax
2

and
y

=
x
2

+

c

to each student. Display teacher resource:
Investigating
y

=
ax
2

and
y

=
x
2

+

c
, and facilitate a class
discussion on the e
ffects of changes in “a” and
ATTACHMENTS



Teacher Resource
:
Investigating
y

=
ax
2

and
y

=
x
2

+

c

KEY

(
1 per teacher
)



Teacher Resource
:
Investigating
y

=
ax
2

and
y

=
x
2

+

c

(
1 per teacher
)



Handout:
Investigating

y

=
ax
2

and
y

=
x
2

+

c

(
1 per
student
)



Teacher Resource
:

Algebra
1/Mathematics

Unit 0
9
: Possible Lesson 0
1

Suggested Duration: 9

days


©2012
, TESCCC

04/21
/13

page
5

of
9

2012
-
2013 Enhanced Instructional Transition Guide

Mathematics Algebra I Unit 09

Suggested

Day

Suggested
Instructional
Procedure
s

Notes for Teacher

“c”

in the function
2

y ax c
 

as students complete their handout.

2.

Distribute

handout:
Exploration of Parameter Changes on Quadratics

and a set of
map
pencils
to each
student
. This
may

be completed as homework, if necessary.


Exploration of Parameter
Changes on Quadratics

KEY

(
1 per teacher
)



Handout
:

Exploration of
Parameter Changes on
Quadratics

(
1 per student
)


MATERIALS



graphing

calculator (1 per
student)



graphing calculator with
display (1 per
teacher)



map
pencils (1 set per student)


TEACHER NOTE


Handout:
Investigating
y

=
ax
2

and
y

=
x
2

+
c
is teacher led. The
complete description of the
activity is explained in detail on
the handout. Students investigate
transformations of quadratics
based on the parent function
y =

, using their graphing
calculators.


5

Topics:



Quadratic functions



Parameter changes


E
xplore/Explain

3

Students
explore effects of parameter changes on the quadratic parent function.


Instructional Procedures:

1.

Prior to instruction

create card sets:
Got a Parabo
-
lem?

Card Sort Master

by copying on cardstock,
laminating,
cutting out, and placing each set in a plastic zip bag. One card set will be needed for each
group of 3 students.

2.

Facilitate a class discussion to debrief
Exploration of Parameter Changes on Quadratics

to check for
student understanding.

ATTACHMENTS



Teacher Resource
:

Got a
Parabo
-
lem?

Card
Sort
Master

KEY

(
1 per student
)



Handout:
Got a Parabo
-
lem?
Recording Sheet

(
1 per
student
)



Card Set:
Got a Parabo
-
lem?

Card Sort Master

(
1 set per 3
students
)



Teacher Resource

Parabo
-
lems Galore

KEY

(
1 per
teacher
)



Handout:
Parabo
-
lems Galore


Algebra
1/Mathematics

Unit 0
9
: Possible Lesson 0
1

Suggested Duration: 9

days


©2012
, TESCCC

04/21
/13

page
6

of
9

2012
-
2013 Enhanced Instructional Transition Guide

Mathematics Algebra I Unit 09

Suggested

Day

Suggested
Instructional
Procedure
s

Notes for Teacher

Ask:



What does t
he negative sign in front of “a” do to the function?

(
Reflection
across the

x
-
axis
.
)



What would this type of transformation be called?
(
r
eflection
)



What effect does “a” have on the function?
(
The “a” causes it to stretch or compress vertically
.
)



What if th
e absolute value of “a” is greater than 1?
(s
tretches vertically
)



What if the absolute value of “a” is between 0 and 1?

(
c
ompresses vertically
)



What would this type of transformation be called?
(
d
ilation
)



What effect does “c” have on the function?
(
The “c”

causes it
to shift vertically up or down.)



What if the “c” is negative?
(
v
ertical shift down
)

Positive?
(
v
ertical shift up
)



What type of transformation is this shift called?
(
t
ranslation
)

3.

Place students in groups of 3. Distribute handout:
Got a
Parabo
-
lem? Recording Sheet

to each student.
Distribut
e

1
card set
:
Got a Parabo
-
lem?

Card Sort Master

t
o each group of 3
students
.

Instruct
students to

first sort their cards into 4 groups: an equation group; a group with graphs; a domain/range
group; and a verbal description group.

Instruct students to
proceed
with

or
ganizing

their cards into
match
sets
, contain
ing one
card
from each of the groups, veri
fying their matches with other group members
,

and
recording
their answers on their recording sheet.

Allow students time to complete the questions. Monitor
and assess students to check for understanding.

Facilitate a class discussion of student results, cla
rifying
any misconceptions.

4.

Distribute handout:
Parabo
-
lems Galore

to each student.
Instruct students

to work independently

to
complete the handout
. Th
is may
be completed for homework, if necessary.


(
1 per
student
)


MATERIALS



graphing

calculator (1 per
student)



graphing calculator with
display (1 per teacher)



cardstock (4 sheets per 3
students)



scissors (
1 per teacher)



plastic zip bag
(sandwich
size
d
)
(1 per 3 students)


TEACHER NOTE


Encourage students to
use their
graphing calculator to investigate
the transformations on their
cards.


TEACHER NOTE

Students may be confused with
the terminology “stretch” and
“compress.” In their minds
compress may seem to mean
gets narrower and stretch may
seem to mean gets
wider. Use a
diagram of a parabola and
illustrate what happens when a
point is stretched vertically or
compressed horizontally.


6

Topics:



Quadratic functions



Parameter changes



Scatterplots


Elaborate 1

Students
apply parameter changes to fit a quadratic to a scatterplot of a set of data in order to determine a
ATTACHMENTS



Teacher Resource
:

Drop That
Ball

KEY

(
1 per teacher
)



Handout:
Drop That Ball

(
1
per student
)


MATERIALS



graphing

calculator (1 per

Algebra
1/Mathematics

Unit 0
9
: Possible Lesson 0
1

Suggested Duration: 9

days


©2012
, TESCCC

04/21
/13

page
7

of
9

2012
-
2013 Enhanced Instructional Transition Guide

Mathematics Algebra I Unit 09

Suggested

Day

Suggested
Instructional
Procedure
s

Notes for Teacher

function rule for a problem situation.


Instructional Procedures:

1.

Place students in groups of 3. Distribute handout:
Drop That Ball
to each student.
Distribute 1 sheet of
chart paper and 1 set of chart markers to each group.

2.

Instruct students to work with their group to complete
handout:
Drop That Ball
,

and create and post a
display of their results on chart paper using the model below.


Problem

Independent
variable

Dependent
variable

Domain & range

Table

Graph

Questions

answered


3.

Using a round robin setting
,

moving in
groups

around the room, instruct students to compare and contrast
results on all chart displays. Facilitate a class discussion on student observations.


student)



graphing calculator with
display (1 per teacher)



chart paper (1 sheet per 3
students)



chart markers (1 set per 3
students)


STATE RESOURCE

TMT
3

Algebra 1:
Explore/Explain
Phase 2


The Canopy Tour may
be used
to reinforce these
concepts or may be used as
alternative activities.


7


8


Topics:



Applications of quadratic functions



Parameter changes on quadratic functions


Elaborate 2

Students
collect and analyze data to determine a function rule for a problem
situation. Students use the
representations of the data to make predictions and draw conclusions. Students explore parameter changes
on the quadratic parent function.


Instructional Procedures:

1.

Place students in groups of 3. Distribute handout:
Net Gains

and 1 pair of scissors
to each student.
Distribute 6
sheets of centimeter grid paper

and 1 roll of
clear

tape to each group of 3.

2.

Facilitate a class discussion on building nets of prisms and investigating the relationship between the
dimensions and surface

area to discover the quadratic relationship.

Ask:



What is a net?
(
A two
-
dimensional view of a three
-
dimensional shape that shows all the faces and
bases of the prism.
)



How can you find the surface area

using the net
?
(Count or calculate

the area of each

of the

face
s
ATTACHMENTS



Teacher Resource
:
Net Gains

KEY

(
1 per teacher
)



Handout:
Net Gains

(
1 per
student
)



Teacher Resource
:

Transformations: More Than
Meets the Eye

KEY

(
1 per
teacher
)



Handout:
Transformations:
More Than Meets the Eye

(
1
per student
)


MATERIALS



graphing

calculator (1 per
student)



graphing calculator with
display (1 per teacher)



grid paper (
centimeter
)

(6


Algebra
1/Mathematics

Unit 0
9
: Possible Lesson 0
1

Suggested Duration: 9

days


©2012
, TESCCC

04/21
/13

page
8

of
9

2012
-
2013 Enhanced Instructional Transition Guide

Mathematics Algebra I Unit 09

Suggested

Day

Suggested
Instructional
Procedure
s

Notes for Teacher

and bases, and
add the areas.
)



What is the formula for area of a rectangle?
(
Area =

base x height or Area = length x width.
)

3.

Instruct students to work with their group to complete handout:

Net Gains
.
Allow students time to
complete the questions. Monitor and assess students to check for understanding.
Facilitate a class
discussion of student results, clarifying any misconceptions.

4.

Distribute handout:
Transformations: More Than Meets the Eye
to each stu
dent.
Instruct students to
work independently to complete the handout. This may be completed as homework, if necessary.


sheets per 3 students)



scissors (1 per student)



tape
(
clear
)
(1 roll per 3
students)


TEACHER NOTE


For the handout:
Net Gains
,
remind students to tape prisms so
that the grids
appear on
the
outside

of the prism
.


9

Evaluate 1

Instructional Procedures:

1.

Facilitate a class discussion to debrief handout:
Transformations: More Than Meets the Eye

to check
for understanding.

2.

Assess student understanding of related concepts and processes by using the Performance Indicator(s)
aligned to this lesson
.


Performance

Indic
a
tor(s):

Create a mini
-
notebook that explains the effects of changes in “
a
” and “
c
” on the graph of
2
y ax c
 
. The
notebook should include the following:



Graphical representations



Symbolic representations



Verbal descriptions



Transfo
rmations


Reflections (across the
x
-
axis), translations (up and down only), and dilations
(stretches and compressions)

(A.2A; A.9A, A.9B, A.9C, A.9D)


1E; 3H; 5F


Analyze a problem situation that can be represented by a quadratic function such as the
following:




Pablo wants to build a pen for the pig he is raising in his agriculture class. Pablo only has 36 feet of
fencing. What dimensions can Pablo use to give the pen the maximum area?


Create a graphic organizer that includes a table, scatterplot, parent function, and representative function. Use
the representations to determine characteristics of the function, including independent/dependent variables,
domain/range, continuous/discrete
data, increasing/decreasing intervals, vertex, minimum/maximum, axis of
symmetry,
y
-
intercept, and
x
-
intercepts. Explain the meaning of each characteristic in terms of the problem
situation. Justify the pen Pablo should build in order to yield the maximum
area. (A.1D, A,1E; A.2A, A.2B,
ATTACHMENTS



Teacher Resource (optional):
Quadratic Functions

KEY

(1
per teacher)



Handout (optional):
Quadratic
Functions

PI

(1 per student)


MATERIALS



graphing

calculator (1 per
student)


TEACHER NOTE

As an optional assessment tool,
use handout (optional):
Quadratic Functions

PI
.



Algebra
1/Mathematics

Unit 0
9
: Possible Lesson 0
1

Suggested Duration: 9

days


©2012
, TESCCC

04/21
/13

page
9

of
9

2012
-
2013 Enhanced Instructional Transition Guide

Mathematics Algebra I Unit 09

Suggested

Day

Suggested
Instructional
Procedure
s

Notes for Teacher

A.2C, A.2D; A.9A, A.9D)


1C; 3H