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Level 7
Teacher’s Guide
Welcome
Table of Contents
Program Welcome. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Research
Research on Mathematics Intervention . . . . . . . . . . . . . 6
The Need for Intervention . . . . . . . . . . . . . . . . . . . . 7
Response to Intervention in Mathematics . . . . . . . . 8
Components of Effective
Mathematics Interventions . . . . . . . . . . . . . . . 10
High-Yield Strategies for Increasing
Student Achievement. . . . . . . . . . . . . . . . . . . . 12
Using Technology to Improve
Mathematical Learning . . . . . . . . . . . . . . . . . . 13
Using Games to Motivate Struggling
Math Learners . . . . . . . . . . . . . . . . . . . . . . . . . 14
Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Best Practices
Components of Effective Mathematics Intervention
Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Differentiating by Specific Needs. . . . . . . . . . . . . .
Developing Academic Vocabulary. . . . . . . . . . . . . . . . .
Academic Vocabulary. . . . . . . . . . . . . . . . . . . . . . . .
Developing Math Skills Using Concrete
Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Developing Mathematical Problem-Solving Skills. . . .
Why We Teach Problem Solving. . . . . . . . . . . . . . .
Making Connections . . . . . . . . . . . . . . . . . . . . . . . .
A Problem-Solving Framework. . . . . . . . . . . . . . . .
Math in the Real World. . . . . . . . . . . . . . . . . . . . . . . . .
Developing Math Fluency Skills. . . . . . . . . . . . . . . . . .
17
19
19
21
21
31
32
33
35
35
36
39
40
Planning for Intervention
Pacing Plans. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction to Correlations. . . . . . . . . . . . . . . . . .
Standards Correlations. . . . . . . . . . . . . . . . . . . . . . .
Series Scope and Sequence. . . . . . . . . . . . . . . . . . . . . .
© Teacher Created Materials
22
24
24
24
25
28
29
How to Use This Product
Kit Components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Getting Started. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Teaching a Lesson . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Using the Math Fluency Games. . . . . . . . . . . . . . . . . .
How to Organize and Manage Games . . . . . . . . . .
Playing the Math Fluency Game Sets. . . . . . . . . . . . . .
Playing the Digital Math Fluency Games. . . . . . . . . . .
Using the Technology Options. . . . . . . . . . . . . . . . . . .
Lessons
41
45
45
46
49
Lesson 1: Unit Rates with Ratios of Fractions . . . . 61
Lesson 2: Proportional Relationships . . . . . . . . . . . 69
Lesson 3: Identifying the Constant
of Proportionality. . . . . . . . . . . . . . . . . . . . . . . 77
Lesson 4: Representing Proportional Relationships
with Equations . . . . . . . . . . . . . . . . . . . . . . . . . 85
Lesson 5: Interpret Graphs of
Proportional Relationships. . . . . . . . . . . . . . . . 93
Lesson 6: Solve Multi-Step
Ratio/Percent Problems. . . . . . . . . . . . . . . . . 101
Lesson 7: Addition with Rational Numbers . . . . . 109
Lesson 8: Subtracting with Rational Numbers. . . 117
Lesson 9: Solving Problems with
Rational Numbers. . . . . . . . . . . . . . . . . . . . . . 125
Lesson 10: Multiplying with Rational Numbers. . 133
Lesson 11: Dividing with Rational Numbers . . . . 141
Lesson 12: Solving More Problems
with Rational Numbers . . . . . . . . . . . . . . . . . 149
Lesson 13: Converting Rational Numbers
to Decimals. . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Lesson 14: Adding, Subtracting, Factoring,
and Expanding Linear Expressions. . . . . . . . 165
Lesson 15: Linear Expressions Related
to Properties of Shapes. . . . . . . . . . . . . . . . . . 173
Lesson 16: Solve Word Problems with
Rational Numbers. . . . . . . . . . . . . . . . . . . . . . 181
Lesson 17: Solve Multi-Step Problems
Using Estimation . . . . . . . . . . . . . . . . . . . . . . 189
Lesson 18: Solve Equations Containing
One Variable. . . . . . . . . . . . . . . . . . . . . . . . . . 197
Lesson 19: Writing Inequalities to
Represent Word Problems. . . . . . . . . . . . . . . 205
Lesson 20: Write and Graph Inequalities. . . . . . . 213
Lesson 21: Solving Problems with
Scale Drawings. . . . . . . . . . . . . . . . . . . . . . . . 221
Lesson 22: 2-D Planes in 3-D Figures . . . . . . . . . 229
Lesson 23: Area and Circumference of Circles . . 237
Lesson 24: Solving Problems with Angles. . . . . . . 245
Lesson 25: Solving for Unknown Dimensions . . . 253
Lesson 26: Understanding Random Samples. . . . 261
Lesson 27: Comparing Data . . . . . . . . . . . . . . . . . 269
Lesson 28: Understanding Probability . . . . . . . . . 277
Lesson 29: Calculating Relative Frequency. . . . . 285
Lesson 30: Representing Compound Events. . . . 293
Appendices
Appendix A: References Cited . . . . . . . . . . . . . . . . . . 301
Appendix B: Teacher Glossary. . . . . . . . . . . . . . . . . . . 304
Appendix C: Digital Resources Charts. . . . . . . . . . . . 314
21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
3
HOW TO USE
RESEARCH
THIS PRODUCT
Kit Components
Teacher’s Guide
30 easy-to-use, standards-based lesson plans
3 Digital Math Fluency Games
Focus on mathematical skills and strategies,
and are on the Digital Resources USB Device
Level 7
Level 7
Teacher’s Guide
Student Guided Practice Book
Full-color student activities
Level 7
Digital Resources
• PDFs of all student materials, game
sets, activity sheets, assessments, etc.
• PDFs of teacher resources
• Digital Math Fluency Games
• Electronic versions of the Pretest,
Posttest, Performance Tasks, and
reporting tools
Assessment Guide
Includes a pretest, posttest, performance
tasks with assessments, and the answer key
for the Student Guided Practice Book
Refocus Mini Lesson PPT
Provide as PowerPoint® and PDF files
Level 7
Level 7
Assessment Guide
3 Math Fluency Game Sets
Include a game board, directions, an answer
key, and game pieces
directions
Skill:
Single-Variable
Equatio
ns
How to Win
Solve equatio
ns as you travel
Setting Up
across the galaxy
to reach Finish!
Time
3(4 + x) = 18
x=?
5(6 + x) = al
55Spaces
x = ? Close encounter!
i10238
Madness! Game
Cards
1
i10238
23083 (i10265)—F
2
ocused Mathematic
s—Meteor Madness!
6(r + 2) = 60
r=?
i10238
Game Set
3
© Teacher Created
9(3 + w) = 54
w=?
i10238
i10238
5
i10238
6
© Teacher Created
Materials
© Teacher Created Materials
Materials
4(8 + y) = 36
y=?
4
p(5 + 8) = 26
p=?
i10238
7(q + 5) = 63
q=?
Switch places
with another
The stars align!
space traveler
Shoot forward
!
two more spaces!
Mayday! Mayday
! Go back to
Start.
m(14 + 6) =
80
m=?
© Teacher Created
Materials
23083 (i10238)—F
ocused Mathematic
s—Meteor
1. Roll the
number
and solve. Followcube and move forward
that many
the instructions
2. If you solve
below for special spaces. Then, draw a
correctly, move
card
spaces.
(Check the
Answer Key.) forward two spaces. If
not, move back
3. The first
player to reach
one space.
Finish wins! i10238
wins.
If time runs
out, the player
closest to Finish
Note: Separat
e used cards
cards during
as you play.
the game.
Shuffle and
reuse them
if you run out
of
Speci
i10238
23083 (i10238)—F
ocused Mathematic
s—Meteor
s! game board
cards
pawns
number cube
scratch paper
and pencils
1. Place the
Answer Key
game board
in the middle
for problem
of all players.
solving.
Distribute paper
2. Shuffle
the cards
and pencils
chooses a pawn and place them facedow
n on the game
and places it
3. Players
on
board. Each
Start.
take turns rolling
player
the number
continues to
cube. The lowest
the left.
roller goes first,
and play
Game
Madness! Game
Cards
Meteor Madnes
2(16 + v) = 48
v=?
7
8
9
21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
31
HOW TO USE
RESEARCH
THIS PRODUCT
Teaching a Lesson
Teacher’s Guide
Each 8-page lesson is organized in a consistent format for ease of use. Teachers may choose to
complete some or all of the lesson activities to best meet the needs of their students. Lesson
materials can be utilized flexibly in a variety of settings. For example, modeling with a small
group, using printed materials with a document camera, or using PDF materials on a digital
platform, such as an interactive whiteboard. Each lesson includes:
Addition wit
h
Rational
Numbers
Learning Obj
ectives
The Numb
er System
• Understand
p + q as the
number located
|q| from p,
in the positiv
a distance
e or negativ
depending
e direction
on whether
q is
Show that a
number and positive or negative.
its opposite
0 (are additiv
have
e
numbers by inverses). Interpret sums a sum of
describing
of rational
real-world
• Solve real-wo
concepts.
rld
involving additio and mathematical proble
ms
n with rationa
Mathematica
l numbers.
l Practices
and Proces
ses
• Make sense
of
solving them. problems and persevere
in
• Construct
viable argume
nts and critiqu
reasoning of
others.
e the
• Model
with mathem
atics.
• an overview page with key information for planning
• key mathematics content standards covered
• key mathematical practices and processes addressed
• an overview providing teacher background or
student misconceptions
Name: ____________________________________________________________
Name:
_____________________________________________
Lesson
Name: _____________________________________________________________________
Name:
_________________________________________________
7
_________
____
Date: _________________________
Date:
_________________
Directions:
Something Rational
Directions: Complete each of
Rational
Complete each
1 Use counters:
the following.
1 Use counters.
of the following.
Date: _________________________
Date:
_________________
Sums
Lesson
7
-4 + 7 = _______________________________________
_____________________
-8 + (-4) = ____________________________________________________________________
2 Use a number
5 + (-7) = ____________________________________________________________________
8 + (-4) = ____________________________________________________________________
-5
Lesson
5
8
1
4
1
2
2
8
3
8
3
4
4
8
5
8
-2
-1
0
1
Directions:
2
3
4
5
7
8
-2
-1
0
______________________
7
8
2
4 Solve: -5 5 5
+4
6
1
2
3
4
6
= _______________________________________
5
Which
________________________________
_____________________
Quick
Choose
all correct
Date: _________________________
Date:
_________________
Check
answers.
equations
are true?
A -4 +
5=8+
(-7)
B 9+
(-6) = -7
+4
C -10
+ (-5) =
8 + (-7)
D 12
+ (-2) =
-6 + (-4)
E
0.50 = _______________________________________
1
6
8
3.5 + (-4.5) = _________________________________________________________________
-3
Name:
Name:
_____________________________________
____________________________
_____________
-3
1
3 Solve: -1.25
+
+ (- 83 ) = ____________________________________________________________________
1
8
7
line: 4 1 + (-3 1
2
2 ) = _______________________________________
-4
2 Use a number line to solve.
_________________________
8 + (-5)
= 14 +
(-11)
At midnight
it was -3
the temperature
degrees.
By 8:00
at 8:00
a.m. the
a.m.? Explain
temperature
your reasoning.
decreased
by 12 degrees.
What is
________________________
________________________ ________________________
3
© Teacher Created
Materials
21208—Focused
48
21208—Focused Mathematics—Student
Guided Practice Book
Mathematics—Student
Guided Practice
Book
49
© Teacher Created Materials
________________________ ________________________
Jennifer
dives into
down another
________________________ _____
the sea
from
8
Where
is she in meters below the top of a cliff
sea level.
relation
_____
that is 15
to sea level?
She then
meters
swims
above sea
Explain
up 5 meters
level. She
your reasoning.
towards
continues
the surface.
________________________
Lesson
Name: _____________________________________________________________________
Name:
_________________________________________________
7
Name: _____________________________________________________________________
Name:
_________________________________________________ Date:
Date:_________________________
_________________
Independent Practice
Lesson
7
Refocus
Directions: Complete
Stock
Real World
Make a
Plan
-5
-4
-3
-2
-1
0
1
Solution
0
A.M. was -12.5 degrees.
What was the temperature
At noon it was 16.5 degrees
at noon? Explain your
warmer.
reasoning.
-7 + (-3) = ____________________________________________________________________
-9 + 4 = ______________________________________________________________________
Look Back
______
Guided Practice Book
Created
Materials
________________________
__________________
and Explain
_______________________________________________________________________
21208—Focused Mathematics—Student
_____
Book
Name:
Name:
________________________
__________________
_____________________
_____________
Date:
Date:
________________________
_________________
ion
_
________________ ________________ ________________
__
_______________________________________________________________________ ______
52
21208—Focused Mathematics—Student Guided Practice Book
Practice
© Teacher
7
________________ ________________ ________________
________________ __
________________ ________________ ________________
__
2 How
modelcan you use
________________ ________________ __
to explain a number
________________
your
line
________________ __
thinking.
to add
rational
________________ ________________
numbers?
__
________________ ________________ ________________
Write
an example
________________ ________________ ________________
and
________________
draw
a visual
________________ ________________ ________________
________________
3 The temperature at 8:00
4
Guided
Lesson
________________ ________________ ________________
________________ ________________ __
________________ ________________ ________________
________________ ________________ __
________________ ________________ ________________
________________ ________________ __
-1
1
© Teacher Created Materials
7
________________ ________________ ________________
________________
________________ ________________ ________________
________________ ________________
2
3
4
0.8 + (-0.5) = _________________________________________________________________
5
5
________________________ _____
Mathematics—Student
Lesson
Reflect
-3.5 + 6.5 = ___________________________________________________________________
-8 + (-2) = ____________________________________________________________________
-5 + 3 = ______________________________________________________________________
21208—Focused
1 What
did you
________________
learn
about
adding
________________ ________________
rational
numbers?
________________ ________________ ________________
2 Use a number line.
2
3
________________________ ________________________
50
Date: _________________________
Date:
_________________
Market
For the
week of
July 22,
certain
stock: -2
the
The stock
Monday; following day-to-day
+4 Tuesday;
changes
Write an began the week
-8 Wednesday;
in the stock
equation
at
and find 78 points. How
market
+2 1
were recorded
the solution.
many points 2 Thursday; 1
did it finish -3 4 Friday.
for a
with at
the end
Unpack
of the week?
the Problem
-5 + 9 = ______________________________________________________________________
2 + (-10) = ___________________________________________________________________
8 + (-8) = ____________________________________________________________________
6 + (-7) = ____________________________________________________________________
________________________ ________________________
___________________________
__________________
Math in the
each of the following.
1 Use counters.
Directions: Complete each of the following by using counters.
1
Name: __________________________________________
Name:
_______________________________
Date: _________________________
Date:
_________________
________________ ________________ __
© Teacher
Created
________________ __
Materials
© Teacher Created Materials
21208—Focused
__
Mathematics—Student
Guided Practice
51
Book
53
54
7
Student Guided
Practice
Book (pages
48–54)
• Math Fluenc
y
Game Sets
• Digital Math
Fluency Gam
es
• Counters
(filename:
counters.pdf)
• Number
Lines
(filename:
numline.pdf)
Progress Mon
itor ing
The Studen
t Guided Practic
to formally
e Book pages
and
the concepts. informally assess studen below can be used
t understanding
of
Lesso n
Materials
•
Teacher
Backgroun
d
This lesson
involves adding
rational numbe
rs, numbers
that can be
written in the
of a ratio or
form
fraction. Studen
should have
ts
some backgr
ound
knowledge
of adding fraction
decimals, and
s,
mixed numbe
using a numbe
rs
r
lesson, studen line. In this
ts will add rationa
numbers using
l
a number line. counters and
Simply
giving
students a series
rational numbe of steps to add
rs does not
conceptual
understanding build
visual models
. Using
will
students unders help
meaning behindtand the
adding
positive and
negative
numbers.
21208—Focused
Mathematics—Student
© Teacher Created
Materials
Guided
Practice
Book
© Teacher
21199—Focused
Created
Materials
Mathematics
Intervention
Level 7—Teac
her’s Guide
109
Add ition with
Rational Numbers
Lesson
7
Warm-Up
min.
1. Prior to the lesson,
3 (-3)
write the following
problems on the
(cont.)
board:
10 (-10)
-12 (12)
1
4 12 (-4 2 )
show how you know.”
-2.75 (2.75)
a number line to
on the number
each integer? Use
distance from 0
the opposite of
write the
opposite is the same
2. Ask, “What is
Then, have them
that an integer’s
Remind students
line for each problem.
draw a number
line. Have students
Students
number.
lines and solutions.
opposite of each
to share their number
to come to the board or disagree with the presenter.
3. Ask students
whether they agree
should confirm
• a Warm-Up activity to build students’ recall of important
mathematical concepts
• a whole-class Language and Vocabulary activity
• time markers to indicate the approximate time for instruction
Vocabularyy termsmin.
on the board:
Language and
vocabular
write the following
negative numbers
1. Prior to the lesson,
positive numbers
integers
types of
rational numbers
relates the different rational
are
make a chart that
and rational numbers,it integers. Tell students that integers numbers.
2. For integers
and negative whole
a circle and label
numbers. Draw
are all the positive
specifically they
circle.
numbers, but more
write them in the
give examples and
numbers. Remind
Ask students to
and label it rational
fractions. Tell
the integers circle
as
around
written
be
circle
that can
Ask
3. Make a larger
numbers are numbers
are rational numbers. the
in
students that rational integers they wrote in the circle
did not include
of the
numbers that they
look
students that all
of examples of rational rational circle. The chart should
the
students to think
write them inside
integers circle and
something like this:
rational
numbers
1
2
-1.7
5
integers
17
-1
- 32
23
-6
0.65
refer
4. Have students
ations during the
to these represent
lesson.
Materials
© Teacher Created
Guide
Level 7—Teacher’s
Mathematics Intervention
21199—Focused
110
Les son
LessonAddi7tion with
Lesson
nal Numbers (cont.)
A
AdditRatio
ion7 wit
Ratio dd ition7 w
h
RatioWhole
-Grou
nal Nu
it
nal N
LessonWhole-Gro
mbpers
Whole-G
umberh
up Le
You (cont.)
Do
We Do
n min. roup Les
sson
son1. Refer students to the Something
1. Ref
s (cont.)
Whole-Group Lesso
Rationaleractivity
I Do
stud sheet
page
Lesso n
Addi7tion with
Rational Numbers
(cont.)
(cont.)
(cont.)
(con
t.)
48). Say, “Let’s look
ent
(Student
page
(cont.)
at another rational
4. Say, “Let’s
49). Pro s to the Rat Guided Practice Book,
counters.” First, read
do another
to help number
ion
problem
videaddition
the problem: -8
al Sumand use our
problem togethe
1.50
the
question:
+ (-4)stud
+ (-0.75) =
= _____.
s acti
ents exp sentence
r.” Write
will address this focus
2. Hav
2. Ask,
“Howthe
many
not whole numbe_____. Ask, “Are we
1. The following lesson
followi
of each
frames vity sheet
lain the
e stud
kind of counters
ng proble
adding
do we
(Studen
numbers?
rs; they are
m onexp
from
entneed?”
Ask, “Will
ir pro
integer
thelain
there
s sha (We only
these rationa
s?”be(No.)
board:
need
any zero
decima
Step 7
cesred
How do you add rational
t Guided
students.
pairs
the
l numbers.)
s. counters.)
of
ing the
“Why
counters?
numbe
read
theirnot?”
not?”
Why orrewhy
boardl and
rs.”it to
voc
Say,are
on the
(They
four
Pra
Have studen the counters
solution
negative.)
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(No, because all of of the We
abulary
equal
are ir rea
“What is the
use a numbe
parts
soning,
write the focus question
ts draw a numbe
Do sec ctice Boo
of the lesson.
endbetwee
ter(-12)
the
r line tosum?”
2. You may wish to
n the whole 3. Read
ms.
numbeat
k,
question
remind s and reason
tion of
r
add
focus
line
the
the
r
line
second
from
revisit
numbe
will
to
problem: 5-2+to
ing. If
the less
rs.
them
(-7)
2 divided
Explain that you
= _____.
students to
the right and add the numbers. Studen
Ask students to work
build theAsk
to use
into
student
on
problem
label 1.5. To
usingexplain
with a partner
ts should
their how to use
the sen
s hav
to
the numbe
Ask,
subtract a negativ
to start at zero,counters.
need?” (fivesay
the“How many of each kind
r line.
seven red) then
of counter do we tence fram e difficulty
line.
e numbeyellow,
“How
numbe
a number
move
many
look like this:usingThe
and
es and
r you
to zero pairs can you make?”
counters
landthe
(-2)r, move 0.75 spaces
numbers
on sum?”
I Do
(five) “What is
is the solutio
going to add rational
Revisit to the left on
of a ratio, or fraction.”
form
n.
the
in
The
1. Say, “Today we are
visual
expressed
4.
Read the next problem: the model
the focu
numbers that can be
should
8 + y can
= _____.
s questudents
Rational numbers are
the problem using their or (-4)
The red counters represent
stion: to work independent
use to Ask
counters.
integers using counters.
How
a
ly
to
add
add
build
to
Ask,
num
Write
“How
going
are
many
integers.”
rationaof eachdokind
need?” (eight
ber
–0.75 yellow,
youof
2. Say, “First, we
to red)
represent positive
counter do we
four
l
add rati
add
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of students.
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visual model of a number
Focus
• a whole-class section focusing on the key concept/skill
being taught
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Whole‑Group lesson section
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© Teacher Created Materials
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© Teacher Created Materials
21199—
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Mathem
21199—Focuse
atics Inter d Mathematics
Intervention Level 7—Teacher’s
vention
Guide
Level
7—Teach
er’s Guid
e
111
Level 7—Teacher’s Guide
Mathematics
d Mathematics Intervention
21199—Focuse
Intervent
ion Level 7—Teac
her’s Guide
© Teacher Created
Materials
113
© Teac
her Crea
ted Mat
Pacing Plans
plann ing for
interv ention
erials
planning for
intervention
When planning the pacing of a curriculum
program, analyze student data to determine
ine
determOnce
standards on which
t datatotofocus.
a pacing plan is selected or created
analyzeofstuden
based on known
on known
the students
based
program, needs
and/or
the results
d or created
of a curriculum
lessons that of the Pretest, teachers can focus on the lessons that
thewhich
correlate
on for
with
items
ng the pacing Once a pacing plan is selecte
focus
students did not demonstrate mastery.
canthe
When planni
t, teachers
The Pretest is
Pretest is
to determine
the Pretesdesigned
which to focus.
which
conceptstsstudents have already mastered
mastery. The
the results of
standards on
and which concepts
which concep
need tostrate
bemaster
mastered.
ts did not demon
students and/or
ed andTeachers
use this information to choose which
y
cover
which studen
needs of the
alread
to can
lessons to cover
and
which lessons
lessons
the items for
to skip.
which
Even may
ts students have
afterstill
making
these data-driven decisions, teachers
choose
concep
to
correlate with
rs
ationto accelerate orns,
may still
teache the curriculum
determine which rs can use this informhave
in order to meet the needs of the students
riven decisio decelerate
designed to
students in
data-d
ed. Teache
classes. The
in
g thesetheir
following
needs of the
master
makin
are
a
the
a
be
few
easy
to
ways
after
meet
to
change
within
need
the pace of the curriculum within a
order to setting.
to skip. Even
curriculum
lum in whole-class
pace of the
and which lessons or decelerate the curricu
to change the
ways
rate
easy
Ways to Accelerate the Curriculum:
have to accele
ng are a few
. The followi
their classes
• Certain mathematical concepts may
setting.
the
whole-class
ts. If this is come more easily to some students. If this is the
case, allow
lessstuden
time for
the practice
to some
to the
and application of those skills and move
the Curriculum:
on
easily
rate
move
more
on to the
next lesson
inskills and
Ways to Accele
ts may come
tion of those the program.
matical concep
and applica
• Skip those lessons or concepts for
• Certain mathe time for the practice
mastery on which students have demonstrated mastery on
the Pretest.
case, allow less program.
demonstrated
the
students have
next lesson in
ts for which • Reduce the number of activitiesd that students
complete in the Student Guided
lessons or concep
t Guide
Practice Book.
• Skip those
in the Studen
ts complete
the Pretest.
es that studen
Ways to Decelerate the Curriculum:
number of activiti
• Reduce the
• If the concepts in a particular lesson
are very challenging to the students,
Practice Book.
ts, allow more
allow more
ulum:
time for ging
the studenof
eachtocomponent
the
lesson: modeling, guided practice, independent
ndent
challen
erate the Curric
verypractice,
and application
practice, indepe
Ways to Decel
lar lesson are
games and activities.
ing, guided
ts in a particu
model
Use more pair or group activities to
• If the concep component of the lesson: • es.
allow students to learn from one another
while
one another
while
time for each
theirfrom
games and activiti reinforcing
understanding
of the concepts.
ts to learn
application
allow• studen
practice, and
Review Quick Check pages with students
activities to
and have them resolve incorrect items.
ct items.
pair or group
the concepts.
resolve incorre
• Use more
tanding of The
them
unders
have
following
their
and
pacing plans show three options
ts
reinforcing
for using this complete kit. Teachers
with studen
Teachers should
customize these pacing
should
Check pages
ete kit.
plans
according to their students’ needs.
compl
• Review Quick
s for using this
show three option students’ needs.
pacing plans
to their
The following pacing plans according Option Instructional Time
Frequency
Material Notes
Notes
customize these
Option 1 6 Mater
weeks ial
(2 hours/day)
Daily d
30 lessons All lessons covered
Frequency
All lessons covere
ctional Time
30 lessons
Option 2 4 weeks
Option Instru
(2 hours/day)
Daily
Dailycovered
20 lessons 20 key lessons covered
day)
key lessons
weeks (2 hours/
lessons 20
Option 1 6
Option 3 24 20
weeks (60 min./day) Twice a week
Daily
covered 24 lessons 24 key lessons covered
day)
key lessons
weeks (2 hours/
lessons 24
Option 2 4
a week 24
Twice
ay) Note: To further adapt the program
to recom
instructional
mendedtime frames, it is highly recommended
weeks (60 min./d
that teachers give the Pretest
Option 3 24
, it is highly
rds19–30)
Guide,
standa
pages
time frames (Assessment
to determine which standards
tionalnot
ine which
students
instruchave
mastered.
m to
Teachers cantothen
e their
19–30) to determ
use
the Pretest Item Analysis to analyze
, pages
is analyz
r adapt the progra students’
Guide
Analys
their
results
andPretes
selectt Item
lessons
to target.
Note: To furthe the Pretest (Assessment
then use the
give
Teachers can
that teachers
not mastered.
students have and select lessons to target.
students’ results
Pacing Plans
• differentiation strategies to support and extend learning with
the Refocus lesson and Extend Learning activity
• math fluency games that motivate students to develop and
reinforce mastery of basic skills
• a Math in the Real World concept task activity
© Teacher Created Materials
Materials
© Teacher Created
© Teacher Created Materials
21199—Focused
Mathematics
41
Guide
21199—Focused
7—Teacher’sMathematics Intervention Level 7—Teacher’s Guide
Intervention Level
21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
41
33
HOW TO USE
RESEARCH
THIS PRODUCT
Teaching a Lesson
(cont.)
Student Guided Practice Book
Each lesson in the Teacher’s Guide has seven corresponding student pages in the Student
Guided Practice Book:
Lesson
Name: ________________________
Name:
__________________
________________________
__________________
_____________________
_____________ Date:
Date: ________________________
__________________
1
Run Day
Directions: Solve the problem.
• a We Do activity to support the gradual release of
responsibility model
1
Caitlin runs
2
3
mile in
1
6
hour. How far can she run in
one hour?
Write the complex fraction that
What is her rate of speed?
will help you solve this problem.
______________________________
Find the solution. Show your
work.
Caitlin can run __________
______________________________
_________________
miles in one hour.
Rate of speed: _______________
______________________________
2
Jackie runs
1
2
1
6
mile in
____________________
hour. How far can she run in
one hour?
Write the complex fraction that
What is her rate of speed?
will help you solve this problem.
______________________________
Find the solution. Show your
work.
Jackie can run __________ miles
______________________________
_________________
Name: _______________
Name:
___________
_______________
___________
_______________
___________
_______________
___________
_________
_____ Date:
Date: _______________
___________
__________
______
in one hour.
Rate of speed: _______________
______________________________
6
____________________
Directions: Solve
21208—Focused Mathematics—Student
Guided Practice Book
Take a Walk
6
Write the complex
12
hour. How far can
she walk
fraction that will help
____________________
1
in one hour? What
is her rate of spee
d?
you solve this problem.
____________________
Find the solution.
• a You Do activity to facilitate independent practice
____________________
Show your work.
Leticia can walk __________
_________________
miles in one hour.
Rate of speed: __________
____________________
____________________
2 Hal walks 1 mile in 1
3
6 hour. How far can he walk
in
Write the complex
fraction that will help
1
Check
Quick
____________________
• a Quick Check to easily monitor students’ progress
true or false.
True
False
1
Johanna walks one mile in 1 3 hours.
True
False
3 Jake walks mile in hour.
8 mile per hour.
Jake’s speed is 15
True
False
4 Molly walks 37 mile in 12 hour.
Molly walks one mile in
True
False
John’s speed is
2 Johanna walks 14 mile in 13 hour.
3
4
2
5
5
9
mile per hour.
6
7
_________________
____________________
© Teacher Created
Materials
Directions: Based on the information given, choose whether each statement is
1 John walks 23 mile in 56 hour.
speed?
____________________
Show your work.
Hal can walk __________
miles in one hour.
Date:
_________________
Date:_________________________
Rate of speed: __________
____________________
__________
___________
_____
__________________________________________________________
Name:
____________________________________________
Name:
Lesson
_______________
one hour? What is
his rate of
you solve this problem.
____________________
Find the solution.
Lesson
the problem.
© Teacher Created Materials
1 Leticia
walks 5 mile in 5
hour.
_____
21208—Focused Mathematics—
Student
Guided Practice Book
7
Directions: Choose the correct answer.
3
1
How long
5 Mandy is filling a pool with water from a hose. She can fill 10 of the pool in 5 hour.
will it take for the pool to be filled?
A
B
C
D
1
6
hour
2
3
hour
6 hours
10 hours
hour?
6 Kip walks 59 mile in 58 hour. What is Kip’s rate of speed? How far can Kip walk in one
Show your work.
__________
______
Date:
___________
Date:_______________
_________
_____
_______________
___________
_______________
___________
_______________
___________
_______________
___________
Name:
Name:
_
____________________________________________________________________________
© Teacher Created Materials
21208—Focused Mathematics—Student Guided Practice Book
Lesson
1
Refocus
_
____________________________________________________________________________
8
each person
s: Find the rate of speed
walks.
Direction
1 Kim:
1
4
2
3
mile in
hour
Write the complex
fraction that will help
you solve this problem.
• a Refocus activity for students who need more instruction
Show your work.
Kim can walk __________
miles in one hour.
3
4
in
2 Manuel: kilometer
3
5
Write the complex
hour
you solve this problem.
fraction that will help
1
_________________
____________________
____________________
____________________
Find the solution.
_______________
____________________
____________________
Rate of speed: __________
Lesson
_________________
____________________
____________________
____________________
Find the solution.
Show your work.
___________
_________________
Date:_________________________
_____ Date:
__________________________________________________________
Name:
____________________________________________
Name:
in one hour.
__________ kilometers
_______________
Manuel can walk
____________________
____________________
Rate of speed: __________
Independent Practice
Directions: Find the rate of speed and the distance that the student can walk in
one hour.
Student
21208—Focused Mathematics—
Materials
9
Guided Practice Book
© Teacher Created
walk in
1 Dana walks 23 kilometer in 78 hour. What is Dana’s rate of speed? How far can Dana
one hour?
• an Independent Practice page to reinforce mathematical
content taught in the lesson
_
____________________________________________________________________________
_
____________________________________________________________________________
_
____________________________________________________________________________
_
____________________________________________________________________________
one hour?
2 Max walks 14 mile in 12 hour. What is Max’s rate of speed? How far can Max walk in
_
____________________________________________________________________________
_
____________________________________________________________________________
_
____________________________________________________________________________
_
____________________________________________________________________________
3 Mateo walks 25 mile in
hour?
5
6
hour. What is Mateo’s rate of speed? How far can Mateo walk in one
_
____________________________________________________________________________
Name: ____________
Name:
_________
____________
_________
____________
_________
____________
______________________
_________
____________________________________________________________________________ _________
____
Math in the
_
____________________________________________________________________________
Real World
10
• a Math in the Real World concept task for students to apply the
math concept in a real-life scenario
Unpack the Problem
Lesson
1
Make a Plan
Solution
Look Back and
___________
_____
__________________________________________________________
Name:
____________________________________________
Name:
Lesson
1
Reflection
Explain
Date:
_________________
Date:_________________________
© Teacher Created
Materials
21208—Focused
1 What did you learn about solving unit rate problems involving fractions?
______
• a Reflection page for students to share their
mathematical understanding
Date: ____________
Date:
_________
____________
_________
Take a Run
_
____________________________________________________________________________
Robert and Elaina
Materials
© Teacher
the Created
of the way around ran around
high school
21208—Focused Mathematics—Student Guided Practice Book
track during gym
track in 5 minute.
class. Robert
around the track the
6
Later in the day,
ran 1
in 9
Elaina ran 3
2
he can run farther 10 minute. When they meet
after school, Robert 4 of the way
your reasoning. in a minute than Elaina. Is
he correct or incorrect claims that
? Explain
____________________________________________________________________________
Mathematics—Student
Guided Practice
Book
11
______
____________________________________________________________________________
____________________________________________
______________________________________
______
____________________________________________________________________________
______
____________________________________________________________________________
______
____________________________________________________________________________
______
____________________________________________________________________________
______
____________________________________________________________________________
______
____________________________________________________________________________
______
____________________________________________________________________________
2 How
does using visual models help you solve unit rate problems involving fractions?
______
____________________________________________________________________________
______
____________________________________________________________________________
______
____________________________________________________________________________
______
____________________________________________________________________________
______
____________________________________________________________________________
______
____________________________________________________________________________
______
____________________________________________________________________________
______
____________________________________________________________________________
______
____________________________________________________________________________
______
____________________________________________________________________________
12
34
21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
21208—Focused Mathematics—Student Guided Practice Book
© Teacher Created Materials
© Teacher Created Materials
planning for
intervention
Pacing Plans
When planning the pacing of a curriculum program, analyze student data to determine
standards on which to focus. Once a pacing plan is selected or created based on known
needs of the students and/or the results of the Pretest, teachers can focus on the lessons that
correlate with the items for which students did not demonstrate mastery. The Pretest is
designed to determine which concepts students have already mastered and which concepts
need to be mastered. Teachers can use this information to choose which lessons to cover
and which lessons to skip. Even after making these data-driven decisions, teachers may still
have to accelerate or decelerate the curriculum in order to meet the needs of the students in
their classes. The following are a few easy ways to change the pace of the curriculum within a
whole-class setting.
Ways to Accelerate the Curriculum:
• Certain mathematical concepts may come more easily to some students. If this is the
case, allow less time for the practice and application of those skills and move on to the
next lesson in the program.
• Skip those lessons or concepts for which students have demonstrated mastery on
the Pretest.
• Reduce the number of activities that students complete in the Student Guided
Practice Book.
Ways to Decelerate the Curriculum:
• If the concepts in a particular lesson are very challenging to the students, allow more
time for each component of the lesson: modeling, guided practice, independent
practice, and application games and activities.
• Use more pair or group activities to allow students to learn from one another while
reinforcing their understanding of the concepts.
• Review Quick Check pages with students and have them resolve incorrect items.
The following pacing plans show three options for using this complete kit. Teachers should
customize these pacing plans according to their students’ needs.
Option
Instructional Time
Frequency
Material
Notes
Option 1
6 weeks (2 hours/day)
Daily
30 lessons
All lessons covered
Option 2
4 weeks (2 hours/day)
Daily
20 lessons
20 key lessons covered
Option 3
24 weeks (60 min./day)
Twice a week
24 lessons
24 key lessons covered
Note: To further adapt the program to instructional time frames, it is highly recommended
that teachers give the Pretest (Assessment Guide, pages 19–30) to determine which standards
students have not mastered. Teachers can then use the Pretest Item Analysis to analyze their
students’ results and select lessons to target.
© Teacher Created Materials
21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
41
Identifying the Constant
of Proportionality
Learning Objectives
3
Materials
Ratios and Proportional Relationships
Lesson
• Identify the constant of proportionality (unit rate) in
tables, graphs, and equations.
Mathematical Practices and Processes
• Reason abstractly and quantitatively.
• Attend to precision.
• Look for and make use of structure.
•Student Guided Practice
Book (pages 20–26)
• Math Fluency
Game Sets
• Digital Math
Fluency Games
•different-colored
highlighters
Progress Monitoring
The Student Guided Practice Book pages below can be used
to formally and informally assess student understanding of
the concepts.
Name: _____________________________________________________________________
Date: _________________________
Teacher
Background
Lesson
3
Proportionally Constant
Directions: Find the constant of proportionality for the table, graph, and equations.
___________________________
____________________________
Name: ______________
Constantly
3
the constant
Directions: Find
1
of proportionality
for the table, graph,
Apples
4
24
5
30
6
36
4
20
Apples/Pie
7
42
8
48
Lesson
Cars/Lot
25
6
30
7
35
8
40
_______________
14
21
12
18
10
15
8
12
6
__________________
6
3
3
4
5
6
7
8
9
10
3
4
9
8
7
6
5
10
A
B
x
A
B
for each equation?
21208—Focused Mathematics—Student Guided Practice Book
21
B
Guided Practice Book
22
_________________________________
Name: ____________________________________
Date:
3Directions:
1
=
A. students in one class
amount
fed given the
of
of the
be
Complete each
many dogs can
employees how
a table to tell ionality?
kennel set up
t of proport
A local dog
y:
is the constan
Proportionalit
Constant of
dog food. What
_______
____________
____________
A.
Dogs/Food
the table.
Dogs
equation for
Dog food (lb.)
B. Write an
_______
10
5
____________
____________
20
10
30
15
40
20
50
25
to give to friends
12
9
6
3
21
5
35
7
49
9
63
Students/Class
B
6
y
24
4
21
28
y=x
C
for the following table
8
32
one week’s time.
Date: _________________________
Unpack the Problem
5
6
Make a Plan
y party.
y
of
A. What is the constant
the graph?
18
12
40
48
14
12
9
_____
6
1
2
3
4
5
6
dent Guided Practice Book
21208—Focused Mathematics—Stu
24
Practice Book
dent Guided
21208—Focused Mathematics—Stu
dent Guided Practice Book
© Teacher Created Materials
B
23
© Teacher Created Materials
________
table help
__
________
________
__
________
________
__
________
________
__
________
________
__
________
________
__
________
__
________
________
__
________
________
________
__
________
________
________
________
________
________
________
________
________
________
________
__
__
__
__
________
________
________
________
________
________
________
________
________
________
s?
________
________
________
________
________
________
________
________
________
________
________
________
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using ratio
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problems
?
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you solve
purchase
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items to
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26
Materials
© Teacher Created
using a
t which
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Date: ______
____________
_______
n
sions abou
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x
Guests
How does
make deci
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3
© Teacher Created
Materials
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2
Look Back and Explain
64
_____
_____
Reflect
io
help you
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Solution
proportionality of
16
56
__________________
__________________
________
increases.
___________________
________________________
the graph.
B. Write an equation for
___________________
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15
Explain.
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© Teacher Created Materials
Mathematics—Stu
________
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________________________
unit rates
________
________
________
at his birthda
y = 7x
is y = 1 x. Is he correct
4
or incorrect?
10
Guided Practice Book
Name:
How do
________
x
21208—Focused
1
________
as the number of students
7
49
D
__________________
3 Less
on
3
________
the number of library books
6
42
Lesson
MPG
________
________________________
The following graph shows
Which is an
5
35
y = 3x
__________________
21208—Focused Mathematics—
Student
Meagan wants to know
how many miles per gallon
(mpg) her car gets. She
of the gallons of gas she
makes a table to keep track
uses. Assume the table
is proportional. Find the
Write an equation representing
constant of proportionality
.
Use the equation to complete the distance, d, she can drive given any number
of gallons of gas, g.
the table.
Gas
3
5
11
13
Distance
15
99
165
264
330
363
429
Students
3
2
y = 71 x
x
70
________________________
of
of proportionality
the constant
A. What is
Jake’s graph?
_______
____________
____________
____________
the graph.
equation for
_
B. Write an
____________
____________
______
____________
15
Favors
favors he wants
Students
the table.
B. Write an equation for
Library Books
number of party
students/class
10
2
y
at Washington Middle School.
of proportionality.
Classes
Math in the
Real World
table and graph.
of students in each class
the constant
Complete the table to find
following.
18
for each
Find the constant of proportionality
the number
The following table shows
Refocus
Date: _________________________
Independent Practice
Lesso n
_______
__________________
10
30
x
10
__________________
______
__________________
__________________
__________________
Name: _________
1
9
__________________
Name: ____________________________________
_________________________________
4
8
3
__________________
Materials
© Teacher Created
3
3
7
__________________
Student
21208—Focused Mathematics—
graph of the
14
A
© Teacher Created Materials
6
2
y = 13 x
1
y = 2x
Lesson
Jake made a
5
1
7
y = 7x
y = 8x
Directions:
4
Days
Laps
3 Jaxson says that the
equation
proportionality
is the constant of
20
3
the
equation for the table? total number of laps he swam over
Pies
3 What
2
2 Mark made a table
of
3 What is the constant of proportionality for each equation?
2
8
24
_____________
x
Lots
1
6
18
C. d = 9t _________
_____________
D. d = 1 t _________
5
_____________
1
2
25
5
4
12
2
1
30
10
Time
Distance
__________________
4
9
40
B.
16
________________________________
24
45
graphs, and equations.
18
Constant of Proportionality:
27
ality:
ality for the tables,
y
20
30
Date: _________________________
Check
each of the following.
1 Find the constant
of proportion
y
2
15
2
Quick
Directions: Complete
20
1
Name: ______________
____________________________
___________________________
3
A.
5
35
Apples
Cars
cars in one lot = cars/lot
Constant of Proportion
50
Lots
and equations.
Pies
= apples/pie
y
2
the parking lots of Harris Company. Complete the
table to find the constant of proportionality.
Proportional
Complete the
baking apple pies.
Jill made a table for
of proportionality.
table to find the constant
apples in one pie
1 The following table represents the number of cars in
Cars
Lesson
Date: _________________________
__
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__
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__
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________ 25
B
__
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__
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21208—Fo
________
________
cused Mathe
__
________
matics—St
________
udent Guide
______
d Practi
ce Book
In this lesson, students will
use what they know about unit
rates to identify the constant of
proportionality in tables, graphs,
and equations. The unit rate is
found by dividing the numerator
by the denominator of a ratio.
The unit rate is the constant of
proportionality. The constant of
proportionality in an equation is
the coefficient of x (independent
variable). In a table or graph,
the constant of proportionality is
y, when x = 1. Students should
be familiar with how points are
graphed on the coordinate plane.
A point with coordinates (x, y)
is graphed by starting at
the origin (0, 0), moving
x-units right or left,
and moving y-units
up or down.
© Teach
er Create
d Mater
ials
21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
77
Identifying the Constant
of Proportionality (cont.)
Lesson
3
Warm-Up
min.
1.Draw the following table on the board:
x
5
6
7
8
Say, “Find the unit rate using k =
y
x
y
15
18
21
24
for each of the lines in the table.” ( 31 )
2.Remind students that to find the unit rate they must divide the value for y by the value
for x.
3.Some students may think the answer is 13 instead of
that they need to find the unit rate using k = yx .
3
1
. If that happens, remind students
4.Ask students if the table represents a proportional relationship. Students should find
that the unit rate, or constant of proportionality, remains the same for each ordered pair.
Therefore, the table represents a proportional relationship.
Language and Vocabulary
min.
1.At the beginning of the lesson, review the following vocabulary terms:
unit rate
constant of proportionality
proportional relationship
coefficient
independent variable
dependent variable
2.Say, “We discovered that the unit rate for the table in the Warm-Up is 31 . This is also
called the constant of proportionality for the table. The constant of proportionality is
represented by k in an equation for a proportional relationship.
3.Point out that the coefficient will be multiplied by the independent variable to equal the
dependent variable in an equation that represents a proportional relationship. Write the
following on a large sheet of paper and have students copy it into their journals or a sheet
of paper:
unit rate = constant of proportionality = k = coefficient of x
k=
y
x
y = dependent variable
x = independent variable
dependent variable = (constant of proportionality)(independent variable)
y = kx
78
21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
© Teacher Created Materials
Identifying the Constant
of Proportionality (cont.)
Whole-Group Lesson
Lesson
3
min.
Focus
1.The following lesson will address this focus question:
How do you find the constant of proportionality for a table, a graph, and an equation?
2.You may wish to write the focus question on the board and read it aloud to students.
Explain that you will revisit the focus question at the end of the lesson.
I Do
1.Say, “Today we are going to find the constant of proportionality in tables, graphs, and
equations. During the Warm-Up, we reviewed finding the unit rate. The unit rate is
also known as the constant of proportionality.”
2.Say, “The table represents Adam’s walk with his dog on a Saturday afternoon.” Write
the following table on the board:
Time Distance
(hours)
(miles)
1
4
3
4
1
2
1 12
3
4
1 14
1 12
2 14
3 34
4 12
distance/
time = speed
1
2
3
1
4
1
3
4
1
4
1
2
1
2
1
4
3
4
3
4
1
4
1
2
1
2
=
3
4
1
4
÷
=
3
4
×
1
= 1 12 ÷
1
2
=
= 2 14 ÷
3
4
=
3
2
÷
3
9
4
= 3 34 ÷ 1 14 =
= 4 12 ÷ 1 12 =
1
4
1
×
=
1
2
=
3
15
4
1
4
3
×
=
9
2
1
×
×
1
4
5
1
2
3
1
1
2
1
=
3
1
3
1
=
1
1
3
3
2
1
1
1
3
1
=
3
1
3
1
3.Say, “ For this table, we need to find the unit rate of distance per unit of time for each
line of the table. So, distance/time = rate of speed. Remember we will be writing
and simplifying complex fractions. Let’s find the constant of proportionality for each
ordered pair in the table.”
© Teacher Created Materials
21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
79
Identifying the Constant
of Proportionality (cont.)
Lesson
3
Whole-Group Lesson
I Do
(cont.)
(cont.)
4.Ask, “What is the unit rate?” ( 31 ) Write the unit rate 31 in the last column for
each ordered pair. Say, “In each case, the unit rate is 3 miles per hour. When we
want to write an equation for this relationship, this unit rate becomes the constant
of proportionality. In this case, the independent variable represents the distance.
The amount of time it takes to walk a distance depends on the amount of distance
walked.”
5.Say, “The equation we need to use is based on the formula for speed.” Write on
the board dt = s, so d = st. Say, “To write an equation that represents this table, we
can write d = 3t. The distance depends on the rate or constant of proportionality
multiplied by the time.”
6.Display the following graph on the board:
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
y
•
•
•
•
x
12345678910111213141516
7.Say, “Look at this graph. How do we know it is proportional?” (It passes through
the origin and is a straight line.) Say, “We can write an equation to represent this
line. The constant of proportionality is also the steepness or slope of the line. We
find the constant of proportionality by finding the ratio of y to x.” Ask students to
tell you the points on the line as you write them on the board. ((4, 3), (8, 6),
(12, 9), (16, 12)) Say, “The rate is the ratio of y to x. To find the rate or constant of
proportionality, we write fractions and simplify them.” Have students determine
the constant of proportionality for the ordered pairs.
8.Say, “The constant of proportionality is 34 . This means that the value for y is always
3
3
4 times x for this graph.” As you say this, write y = 4 x on the board.
9.Write the equation y = 7x on the board. Ask, “Based on the work we have done so
far, what is the constant of proportionality for this equation?” (7) Ask, “What does
this mean?” (It means that y is 7 times x; or in a graph, every time you move 1 unit
along the x-axis, you move 7 units along the y-axis.)
80
21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
© Teacher Created Materials
Identifying the Constant
of Proportionality (cont.)
Whole-Group Lesson
We Do
Lesson
3
(cont.)
1.Refer students to the Constantly Proportional activity sheet (Student Guided Practice
Book, page 20). Say, “Let’s look at more examples of tables and graphs together.
First, look at the table in Question 1.”
2.Ask, “ How do you find the constant of proportionality for this table?” (Divide the
number of apples by the number of pies.) Have students work with a partner to find
the constant of proportionality and complete the table. Then, have students share
their answers and explain their thinking with the class. Students should have found
that the constant of proportionality is 51 , or 5.
3.Look at Question 2. Ask, “How do you find the constant of proportionality for this
graph?” (Find points along the graph and divide the value of y by the value of x.)
Then, have students share their answers and explain their thinking with the class.
Students should have found that the constant of proportionality is 51 , or 5.
4.Look at Question 3. Ask, “How do we find the constant of proportionality in an
equation?” (We look for the coefficient of the independent variable.) Have students
do a choral response by saying the value of the constant of proportionality on the
count of three. Confirm with students that the solutions are 8 and 12 .
5.Have students explain how they found the constant of proportionality. To help
students explain their reasoning, provide them with the following sentence frames:
• In a table, I can find the constant of proportionality by _____ the value of the
_____ variable by the _____ variable. (dividing; dependent; independent)
• In a graph, I can find the constant of proportionality by _____ the value of ____
by the value of _____. (dividing; y; x)
• In an equation, the constant of proportionality is _____ in the equation y = kx,
and is the _____ of the independent variable. (k; coefficient)
Language Support
Each time you use the terminology constant of proportionality,
coefficient, independent variable, and dependent variable, point
to them in the problem. This will enable students to make
connections by associating a vocabulary term to a mathematical
symbol in an equation.
© Teacher Created Materials
21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
81
Lesson
3
Identifying the Constant
of Proportionality (cont.)
Whole-Group Lesson
(cont.)
You Do1.Refer students to the Proportionally Constant activity sheet (Student Guided Practice
Book, page 21). Encourage students to refer to the process to find the constant of
proportionality and identify the dependent and independent variables for tables,
graphs, and equations.
2.Have students share their solutions and reasoning. If students have difficulty
explaining their reasoning, remind them to use the sentence frames and
vocabulary terms.
Closing the Whole-Group Lesson
Revisit the focus question: How do you find the constant of proportionality for a table, a
graph, and an equation? Ask students to explain how to find the constant of proportionality
in tables, graphs, and equations. Students should explain that in a table or graph, the
constant of proportionality can be found by dividing the value of y (dependent variable) by
the value of x (independent variable). In an equation, the constant of the proportionality is
the coefficient of x (independent variable).
Progress Monitoring
min.
1.Have students complete the Quick Check activity sheet (Student Guided Practice
Book, page 22) to gauge student progress toward mastery of the Learning Objectives.
2.Based on the results of the Quick Check activity sheet and your observations during
the lesson, identify students who may benefit from additional instruction in the
Learning Objectives. These students will be placed into a small group for reteaching.
See instructions on the following page.
82
21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
© Teacher Created Materials
Identifying the Constant
of Proportionality (cont.)
Differentiated Instruction
Lesson
3
min.
Gather students for reteaching. The remaining students will complete the Independent
Practice activity sheet (Student Guided Practice Book, page 24) to reinforce their learning and
then play the Math Fluency Games.
Refocus
PPT
Revisit the focus question for the lesson: How do you find the constant of proportionality for
a table, a graph, and an equation? For students who are struggling to find the constant of
proportionality, review how to find the independent and dependent variables. Have students
highlight the dependent variable in one color, and the independent variable in another
color. Remind students that the dependent variable depends on the independent variable.
For example, the number of highlighters needed for the class depends on the number of
students. Ask students to determine the dependent variable (number of highlighters) and
the independent variable (number of students). Have students look at Question 1. Ask them
to identify the independent and dependent variables and highlight them in different colors.
Finally, support students as they complete Question 1 on the Refocus activity sheet (Student
Guided Practice Book, page 23), and then have them solve Question 2 independently.
Math Fluency Games
Math Fluency Game Sets
Digital Math Fluency Games
Extend Learning
Refer students to the Lesson 3 Extend Learning Task (filename: extendtask3.pdf). Tell
them that they will be finding the constant of proportionality by looking at a table.
Some of the information is missing in the table. Ask students how they can determine
the constant of proportionality. Students should explain that they can find the constant
of proportionality by dividing any of the ordered pairs.
© Teacher Created Materials
21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
83
Identifying the Constant
of Proportionality (cont.)
Lesson
3
Math in the Real World
min.
1.Refer students to the Math in the Real World: MPG task (Student Guided Practice Book,
page 25). Have a student read the task aloud. Tell students to explain or summarize the
task to their partner. Have a few students share their summaries.
2.Ask students to think about what information they will need to solve the task and what
the task is asking them to do. Then, have them share with a partner. Ask a few students
to share out. Students need to find the constant of proportionality by dividing the
dependent variable by the independent variable. They should be able to indicate that
the distance depends on the number of gallons of gas in her car. Therefore, the distance
is the dependent variable and the number of gallons of gas is the independent variable.
Have students work in groups of two or three to complete the task.
3.As students are working, circulate and ask focusing, assessing, and advancing questions:
• What is the constant of proportionality?
• How do you use the constant of proportionality to write an equation?
• How do you know your solution is reasonable?
Sentence Frames for Explaining Reasoning
• In this problem, I needed to figure out how many _____ per _____ Meagan’s car gets.
• I found the constant of proportionality by _____.
• I wrote an equation by _____.
4.Observe how students are solving the task, and choose a few groups who solved the task
in different ways to share their solutions and reasoning. Try to have the solutions move
from finding the constant of proportionality to writing the equation. For example, have
students share solutions with a visual representation (a graph of the information) and
then the symbolic representation (equation). Students should find that the constant of
proportionality is 331 after dividing the distance by the number of gallons of gas. The
equation is d = 33g. Make sure students explain their reasoning as they share solutions.
5.As groups are sharing their solution paths, reasoning, and strategies, ask questions:
• Do you agree or disagree with the solution path and reasoning? Why?
• Who can restate ____’s strategy/solution path/reasoning?
• Which solution path makes the most sense to you? Why?
Lesson Reflection
min.
Have students summarize their learning about how to find the constant of proportionality
from tables, graphs, and equations; and provide feedback on any questions they still have
about the content on the Reflection activity sheet (Student Guided Practice Book, page 26).
84
21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
© Teacher Created Materials
Name:______________________________________________________________________Date:__________________________
Lesson
3
Constantly Proportional
Directions: Find the constant of proportionality for the table, graph, and equations.
1 Jill made a table for baking apple pies.
Complete the
table to find the constant of proportionality.
apples in one pie = apples/pie
2
Pies
Apples
4
20
5
25
6
30
7
35
8
40
Apples/Pie
y
50
Constant of Proportionality:
45
_________________________________
40
Apples
35
30
25
20
15
10
5
1
2
3
4
5
6
7
8
9
10
x
Pies
3 What is the constant of proportionality for each equation?
A y = 8x B y = 12 x 20
21208—Focused Mathematics—Student Guided Practice Book
© Teacher Created Materials
Name:______________________________________________________________________Date:__________________________
Lesson
3
Proportionally Constant
Directions: Find the constant of proportionality for the table, graph, and equations.
1 The following table represents the number of cars in
the parking lots of Harris Company. Complete the
table to find the constant of proportionality.
cars in one lot = cars/lot
Lots
Cars
4
24
5
30
6
36
7
42
8
48
Cars/Lot
y
2
30
Constant of Proportionality:
27
________________________________
24
Cars
21
18
15
12
9
6
3
1
2
3
4
5
6
7
8
9
10
x
Lots
3 What is the constant of proportionality for each equation?
A y = 7x B y = 13 x © Teacher Created Materials
21208—Focused Mathematics—Student Guided Practice Book
21
B
Name:______________________________________________________________________Date:__________________________
Lesson
3
Quick
Check
Directions: Complete each of the following.
1 Find the constant of proportionality for the tables, graphs, and equations.
A. y
B.
20
Time
4
6
8
10
Distance
12
18
24
30
18
16
14
________________________________
12
10
C.
d = 9t ______________________
8
D.
d = 15 t ______________________
6
4
2
1
2
3
4
5
6
7
8
9
x
10
2 Mark made a table of the total number of laps he swam over one week’s time.
equation for the table?
Which is an
Days
1
2
3
4
5
6
7
Laps
7
14
21
28
35
42
49
B y = x
A y = 17 x
C y = 3x
3 Jaxson says that the equation for the following table is y =
1
4 x.
D y = 7x
Is he correct or incorrect? Explain.
x
6
8
10
12
14
16
y
24
32
40
48
56
64
______________________________________________________________________________
______________________________________________________________________________
22
21208—Focused Mathematics—Student Guided Practice Book
© Teacher Created Materials
Name:______________________________________________________________________Date:__________________________
Lesson
Refocus
3
Directions: Complete each of the following.
1 A local dog kennel set up a table to tell employees how many dogs can be fed given the amount of
dog food. What is the constant of proportionality?
A.
Dog food (lb.)
Dogs
5
10
10
20
15
30
20
40
25
50
Dogs/Food
Constant of Proportionality:
________________________________
B.
Write an equation for the table.
________________________________
2 Jake made a graph of the number of party favors he wants to give to friends at his birthday party.
y
A.
What is the constant of proportionality of
Jake’s graph?
18
Favors
15
____________________________________________
12
9
B.
Write an equation for the graph.
6
____________________________________________
3
1
2
3
4
5
6
x
Guests
© Teacher Created Materials
21208—Focused Mathematics—Student Guided Practice Book
23
B
Name:______________________________________________________________________Date:__________________________
Lesson
3
Independent Practice
Directions: Find the constant of proportionality for each table and graph.
1 The following table shows the number of students in each class at Washington Middle School.
Complete the table to find the constant of proportionality.
A.
students in one class = students/class
Classes
Students
3
21
5
35
7
49
9
63
10
70
Students/Class
B.
Write an equation for the table.
______________________________________________________________________________
2 The following graph shows the number of library books as the number of students increases.
y
A.
What is the constant of proportionality of
the graph?
Library Books
18
15
____________________________________________
12
9
B.
Write an equation for the graph.
6
____________________________________________
3
1
2
3
4
5
6
x
Students
24
21208—Focused Mathematics—Student Guided Practice Book
© Teacher Created Materials
Name:______________________________________________________________________Date:__________________________
Math in the
Real World
Lesson
3
MPG
Meagan wants to know how many miles per gallon (mpg) her car gets. She makes a table to keep track
of the gallons of gas she uses. Assume the table is proportional. Find the constant of proportionality.
Write an equation representing the distance, d, she can drive given any number of gallons of gas, g.
Use the equation to complete the table.
Gas
3
5
Distance
99
165
Unpack the Problem
264
330
11
13
363
429
15
Make a Plan
Solution
Look Back and Explain
© Teacher Created Materials
21208—Focused Mathematics—Student Guided Practice Book
25
B
Lesson
3
Name:______________________________________________________________________Date:__________________________
Reflection
1 How do unit rates help you make decisions about which items to purchase?
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
2 How does using a table help you solve problems using ratios?
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
26
21208—Focused Mathematics—Student Guided Practice Book
© Teacher Created Materials
Level 7
Assessment Guide
Table of Contents
Research. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Research on Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Why Assessment Is Important. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Understanding Math Content and Practices for Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Types of Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Using the Assessment Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Pretest and Posttest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Pretest and Posttest Item Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Using the Electronic Assessments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Teacher Interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Reporting Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Types of Assessments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Performance Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
How to Use the Performance Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Pretest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Posttest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Performance Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Appendix A: References Cited. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Appendix B: Answer Keys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Pretest Answer Key. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Posttest Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Performance Task Rubrics and Answer Keys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Student Guided Practice Book Answer Key. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Appendix C: Assessment Resources Chart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
© Teacher Created Materials
22205—Focused Mathematics—Assessment Guide
3
Name:______________________________________ Date:___________________
Pretest
1.Carl keeps in shape by cycling. He
bikes 9 miles every 34 hour. How far
will Carl ride in 1 hour?
A6 miles
B10 miles
C12 miles
D14 miles
3
4
1
2
3.The Perez family drove from their
home to Washington, DC. The table
shows the number of miles driven
each day. Which best describes the
relationship between the number of
miles driven and the number of days?
Days
Miles
1
2
1
675
2
1,350
3
1,900
4
2,525
AIt is a proportional relationship
because they drove the same
number of miles each day.
BIt is a proportional relationship
because they drove about the same
number of miles each day.
CIt is not a proportional relationship
because they did not drive the
same number of miles each day.
DIt is not a proportional relationship
2.Jessica runs 12 mile in 16 hour. How far
can she run in one hour?
because they are comparing more
than one quantity: miles and days.
A2 miles
B3 miles
C4 miles
D12 miles
© Teacher Created Materials
22205—Focused Mathematics—Assessment Guide
Go On
19
Name:______________________________________ Date:___________________
Performance Task 1:
Competition Concession Stand
Part A
Belleview School is hosting the winter Belleview Band Competition. Mr. Jamison’s
seventh-grade class has volunteered to make the food and refreshments for the concession
stand.
1.Last year, the stand sold
1
2
gallon of punch in 15 minutes. Find the hourly unit rate.
________________________________________________________________________
2. Mr. Jamison’s daughter is in the high school band. At a recent parade, the high school
band marched 34 of a mile in 30 minutes.
A
How fast did they march in one hour?
_____________________________________________________________________
B
If they marched for 2 12 hours, how many miles did they march?
_____________________________________________________________________
© Teacher Created Materials
22205—Focused Mathematics—Assessment Guide
43