glencoe math triumphs foundations for algebra 1 (teacher handbook)

Contents in Brief

Chapter 1

Numbers and Operations

Chapter 2

Decimals

Chapter 3

Fractions and Mixed Numbers

Chapter 4

Real Numbers

Chapter 5

Measurement and Geometry

Chapter 6

Probability and Statistics

iii


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Table of Contents
Vertical Alignment
Welcome to Math Connects . . . . . . . . . . . . . . . . . . . . . . . . . . T4
An overview of the vertical alignment of Macmillan/McGraw-Hill’s
and Glencoe’s PreK–12 mathematics programs

Implementing Intensive Intervention . . . . . . . . . . . . . . . . . . . . . T6

Program Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T7
Balanced approach of concepts, skills, and problem solving

Assessment
Comprehensive Assessment System . . . . . . . . . . . . . . . . . . . . . T10
Diagnostic, Formative, and Summative assessments linked to
Data-Driven Decision Making

Table of Contents T3

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Welcome to

o
n
C
n
h
ects
t
a
M
Concepts • Skills • Problem Solving
The only true vertically aligned PreK–12 Mathematics Curriculum
Math Connects offers three dimensions of vertical alignment.
1 Content Design
Vertical content alignment is a process
that ensures you and your students

experience an articulated, coherent
sequence of content from grade level to
grade level. This provides you with the
assurance that content is introduced,
reinforced, and assessed at appropriate
times in the series, eliminating gaps and
unnecessary duplication. You are able to
target your instruction to student needs
because you are not teaching content
intended to be covered later or that
students have previously mastered.

TIER

3

2 Instructional Design
Our strong vertical alignment in
instructional approach from
PreKindergarten through Algebra 2
provides a smooth transition for students
from elementary to middle school to
high school. Our common vocabulary,
technology, manipulatives, lesson planning,
and Data-Driven Decision Making reduce
the confusion students often encounter
when transitioning between grade levels
without this built-in articulation.

1

TIER

2

The student pages of Math Connects
have a consistent visual design from
grade to grade. This aids students’
transition from elementary school to
middle school and from middle school
to Algebra 1. Students are more likely to
succeed when they are already familiar
with how to navigate student pages.

Intensive
Intervention

PreK-2

TIER

3 Visual Design

Daily Intervention
Strategic Intervention

T4 Teacher Handbook

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3–5


5 Keys to Success
1 Backmapping

3 Ongoing Assessment

According to College Board research, about 80% of students
who successfully complete Algebra 1 and Geometry by 10th
grade attend and succeed in college. (Changing the Odds:
Factors Increasing Access to College, 1990) Math Connects
was conceived and developed by backmapping with the final
result in mind—student success in Algebra 1 and beyond.

Math Connects includes diagnostic, formative, and summative
assessment; data-driven instruction; intervention options; and
performance tracking, as well as remediation, acceleration, and
enrichment tools throughout the program.

4 Intervention and Differentiated Instruction
2 Balanced, In-Depth Content

A three-tiered Response To Intervention (RTI) is provided.

Math Connects was developed to specifically target the skills
and topics that give students the most difficulty, such as
Problem Solving, in each grade span.
Grades K–2
1.

2.
3.
4.
5.
6.

Problem Solving
Money
Time
Measurement
Fractions
Computation

Grades 3–5
1.
2.
3.
4.
5.
6.

Grades 6–8
1.
2.
3.
4.
5.

Fractions
Problem Solving

Measurement
Algebra
Computation

Daily Intervention Reteach masters and
Alternative Strategy suggestions address concepts from
a different modality or learning style.
2 Strategic Intervention Teachers can use the
myriad of intervention tips and ancillary materials, such
as the Strategic Intervention Guide (1–5) and Study
Guide and Intervention (6–8).
TIER

3

Intensive Intervention For students who are
two or more years below grade level, Math Triumphs
provides step-by-step instruction, vocabulary support, and
data-driven decision making to help students succeed.

Problem Solving
Fractions
Algebra
Geometry
Computation
Probability

5 Professional Development

— K–12 Math Market Analysis Survey, Open Book Publishing, 2006

6–8

1

TIER

Problem Solving
Fractions
Measurement
Decimals
Time
Algebra

Grades 9–12
1.
2.
3.
4.
5.
6.

TIER

Math Connects includes many opportunities for teacher
professional development. Additional learning opportunities in
various formats—video, online, and on-site instruction—are fully
aligned and articulated from Kindergarten through Algebra 2.

Pre-Algebra and Algebra 1

Geometry and Algebra 2

Welcome T5

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Implementing
Intensive Intervention
TIER

Intensive Intervention
3 Data-Driven
Ongoing assessment aids the teacher in student placement, progress monitoring, and exit.
Instructional Design
1

2

3

Classroom Implementation

Diagnose and Prescribe
• Diagnostic and Placement
Test
• Online Readiness Quiz
• Chapter Preview
• Assessment Masters:

Chapter Pretest

Teacher prepares individual or
group intervention plan(s).

Teach and Practice
• Student Study Guide
• Teacher Edition strategies
• Manipulatives

Teacher modifies instruction based on
results of formative assessments.

Advance and Exit
• Progress Check
• Assessment Masters:
Chapter Test

Foundational Skills

Test success indicates that a student can
progress to another Math Triumphs chapter
(if needed) or exit the intervention program.

Algebra 1

for

Number and Operations

Geometry and Measurement

Data Analysis

T6 Teacher Handbook

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Program Organization

Program Organization
Provide Personalized Instruction
Consumable student study guides and minimal preparation requirements allow for flexibility and
personalized instruction in any setting.
• After school

• Summer school

• Before school

• Intersession

• Tutoring

• Pull-out/Resource room

s
Add Fraction

Lesson

3-8

ncept
KEY Co
To add

Y
VOCABULAR

inators , add

h like denom
fractions wit

ors.
the numerat

=

+

_4

_3

=

_7

5

_

2
= 15

+
5
rite the
inators , rew
like denom
e the least
ns with un
ctio
the same. Us
fra
To add
inator .
inators are
om
om
den
den
n
ir
the
commo
fractions so

d the least
ltiple to fin
common mu
5

_5

+

nators
like denomi
rs that are
denominato
the same
denominators
least common
(LCD)
mon
the least com
the
multiple of
rs of two
denominato
tions
frac
e
mor
or
nators
unlike denomi

rs that are
denominato
e
not the sam

Vocabulary
Vocabulary helps students identify terms presented in
the lesson.

Key Concepts

_3

Key Concepts introduce and break mathematics into
conceptual steps. Multiple representations demonstrate
the skills being presented.

4

Hill Compani

es, Inc.

6

10
_

+

7
19 = 1_
9 =_
_
12
12
12

6 and
The LCD of
4 is 12.

per
to an impro
xed number
ms,
vert each mi
ition proble
numbers, con
ns. For all add
To add mixed
proper fractio st form.
im
the
n add
in simple
n
itte
fraction, the
wr

answers are
be sure the

Copyright

© Glencoe/

McGraw-Hill,

a division of

The McGraw-

12

Ex am pl e

s
Add Fraction
Lesson 3-8

2

Circle all set
s in which
the number
-13 belong
s.

GO ON

119

counting
numbers
1. Counting
-13 is not
2. Whole nu

whole
numbers

numbers are
a counting

YOUR TURN
!
Circle all set
s in which
the number
41 belongs.

integers

{1, 2, 3 ...}.

number.

mbers are {0,

1, 2, 3 ...}.

-13 is not
a whole num
ber.
3. Integers
are {..., -3,
-2, -1, 0, 1,
2, 3 ...}.
-13 is an inte
ger.

Examples

41

egers.

Inc.
Companies,
McGraw-Hill
Hill, a divis
ion of The
Copyright
© Glencoe/
McGraw-

}.

a counting
number.
mbers are {

41
3. Integers
41

}.

a whole nu

mber.

are {

an integer.

.

3

Name the opp

osite of -7

.
1. Use a nu
mber line. Beg
in at zero and
move left 7
units. Plot a
point at -7

.
2. Count the
same numb
er of units to
right of zer
the
o. Plot and
label the poi
nt.
7 units
7 units

-8-7-6 -54-3 -2-1

3. The opposi

integers

{

2. Whole nu

4. Circle

Ex am pl e

whole
numbers
numbers are

}.

4. Circle int

Fully worked-out Examples enable students and
parents to see how to solve problems step by step.
Your Turn! gives students an opportunity to
practice skills immediately.

counting
numbers
1. Counting

0 1 2 3
4 5 6 7
8

te of -7 is 7.

YOUR TURN
!
Name the opp
osite of 14
.
1. Use a nu
mber line. Beg
in at zero and
move
units. Plot a
point

at
.
2. Count the
same numb
er of units to
the
of zero. Plo
t and label
the point.

-16-14-12-10-8
-6 -4

3. The opposi

-2 0 2 4
6 8 10 12
14 16

te of 14 is

.

GO ON
Lesson 1-1
Number Gro
ups

5

Program Organization T7

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Guided Prac
tice
Use >, <, or

Guided Practice

1

2.54

= to compar

e each pair

2.45

of numbers.
2

Guided Practice exercises provide computational practice.
They can be used as formative assessment to monitor student
progress and guide your instruction.

0

3

es, Inc.

4

2

2.5

0.185

0.189

3
-1 -.5

the digits the

0

.5

1

1.5

2

same?

After compar
ing the digits,
which place
value helped
inequality sym
decide which
bol to use?
11.12
11.112

Step by Ste
p Practice
5

Order the nu
mbers -2.65
, 1.56, -0.25
least to gre
6, -2.15, and
atest.
0.6

15 from

Step 1 For
the negativ
e numbers,

a division of

Copyright

1.5

463.066
In which pla
ce values are

and
left of

is to the left

on a number
.

<
Step 2 For
the positive

© Glencoe/

McGraw-Hill,

1

After compar
ing the digits,
which place
value helped

inequality sym
decide which
bol to use?

The McGraw-

Hill Compani

Step-by-Step Practice guides students to complete a
computational problem through a series of conceptual steps.
Instructional aids are provided to students in the exercises
that follow.

.5

463.0666

of both

line.

is

<
numbers,

on a number

line.

is to the rig
ht of

<
Step 3 Wr
ite the numb
,

ers from lea
st to greate
st.
,
,

,
GO ON
Lesson 2-2
Compare and
Order Decima
ls

57

tice
lving Prac

-So
p Problem
Step by Ste

He
3
t bike race.
13__ mile dir
to
te entered a 5
RACING Na
w and he had
3 miles when his tire ble
__
en
7
wh
e
ted
comple
8
re left to rac
we
les
mi
ny
e. How ma
quit the rac
he quit?
=
3
__
3
13__ - 7 8 =

5
h step.
Check off eac
key words.
I underlined
Understand:
will
problem, I
the
ve
sol
Plan: To

Solve.
8

.
.
.

answer is
Solve: The

Companies,

Inc.

wer by
cked my ans
Check: I che

McGraw-Hill

epts,
Skills, Conc

McGraw© Glencoe/
Copyright

Hill, a divis

ion of The

Subtract.
5=
7 - __
__
9
8 8

3=
5 - __
__

9 =
6 - ___
__

13

3 =
7 - ___
___

11

2
__
5
9__ - 2 =
6

7

1=
9 - __
___

14

10

16

19

3

14

2
___
1
6__ - 2 15 =
2

17

7

4=
__
1
__
18 5 - 3 5
9

12

12

6

6

3

5

15

Solving
and Problem
10

1=
4 - __
__

12

Step-by-Step Problem-Solving Practice walks the student
through a four-step problem-solving strategy (Understand,
Plan, Solve, Check) that is relevant to the word problem. Aids
help the student break down and visualize what the problem
is asking and how to solve it.

8
__
2
12__ - 8 9 =
3

20

1
__
7
10__ - 3 3 =
9

13
___
3
7__ - 2 20 =
5
GO ON

Lesson 3-9

tions
Subtract Frac

129

T8 Teacher Handbook

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Program Organization

Balance
McGraw-Hill’s Math Triumphs is designed to provide students a balanced approach to
mathematics learning by offering them the opportunity to:
• investigate concepts and build their conceptual understanding;
• review, learn, and practice basic computational and procedural skills; and
• apply mathematics to problem solving in real-world situations.

Skills, Conc

epts,
Find the vol

ume of eac

h figure.

7

and Problem
Solving

22 mm
30 mm

Independent Practice

8
6 cm

9 mm
9 cm

9

0.5 m

6.5 cm
1.5 m

V=
V=

V=

Skill, Concepts, and Problem Solving
provide homework opportunities and
independent practice.

1m

V=

V=
10

sm

b = 3 cm, h
h2 = 12 cm
V=

division of

The McGraw
-Hill Compani

es, Inc.

V=

= 7 cm,

11

rectangular

prism
= 5 yd, w =
2.5 yd,

h = 6 yd
V=

14

triangular pri
sm
b = 4 in., h
= 8.5 in.,
h2 = 7 in.
V=

V=
Solve.
13

12

V=

MOVIES Len
bought a lar
ge box of po
popcorn box
pcorn at the
measured 5
movies. If the
inches by 3
the volume
inches by 9
of Len’s po
inches, what
pcorn box?
was

SWIM Mr
. Armas fill
ed his childr
cubic feet of
en’s inflata
water. The
ble pool wit
length of the
h 40.5
was 4.5 fee
pool was 6
t. Use the for
feet and the
mula h = ___

V__
water in the
width
swimming
· w to find the depth of the
pool.

Copyright

© Glencoe/

McGraw-Hil
l, a

V=

triangular pri

s to which

set
Identify all
16

belongs.
each number
17 4.6

18

GO ON

9
-__
3

Lesson 5-1
The Custom
ary

System

199

67

uary
eived in Jan
e snowfall rec
The averag
m.
WEATHER
simplest for
in
n
ctio
Write as a fra
was 1.2 in.

Solve.

19

20

for the first
ring statistics
.
L Curtis’ sco wn in the table below
BASKETBAL
s to.
are sho
game 4 belong
the season
m
of
fro
es
re
gam
sco
5
t Curtis’
tha
(s)
set
Name the
5
4
3
2

1
71
Game
8 out
4
0.39 of 20 100
Scoring
0.64
9
ics
tist
sta

Vocabulary Check exercises relate directly to the
core vocabulary introduced in each lesson.
Copyright

_

_

© Glencoe/
of The McG
panies, Inc.

raw-Hill Com

140

l, a division

_

A Reflect question requires the student to think
and write about the process of completing a
problem.

McGraw-Hil

ce.

each senten
t completes
lar y word tha
the vocabu
.
eck Write
Ch
y
lar
Vocabu
le of a(n)
is an examp
21 0.125
numbers and
of rational
all examples
are
0, 4, 92
22 -7, 3,

.
mbers, and
als, whole nu
eating decim
ing and rep
ns, terminat
.
23 Fractio
of
all examples
belong.
integers are
ow does not
numbers bel
y one of the
Explain wh
55 . . .
24
75
0.3535535
−
4 out of 12 100
0.2

s
Real Number
Chapter 4

Program Organization T9

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Comprehensive
Assessment System
Data-Driven Decision Making
en t
m
ss
se

and Intervention

1

Sy
st

Diagnostic

em

As

Math Triumphs offers frequent and meaningful assessment of student progress
within the curriculum structure and teacher support materials.

3

Summative

2

Formative

1

Diagnostic

STEP

• Vocabulary Preview
• Assessment Masters:
Chapter Pretest

Get ready for Chapter 1. Review these skills and compare
them with what you will learn in this chapter.
What You Will Learn

You know how to add.

Lesson 1-3

Example: 5 + 4 = 9

You can use a number line to add
integers.

TRY IT!
1

2+5=

2

3+5=

3

4+7=

4

6+6=

-3 + -2 = -5

-6 -5 -4 -3 -2 -1

0

1

2

You know how to subtract.

Lesson 1-4

Example: 12 - 4 = 8

You can use algebra tiles to subtract
integers.

TRY IT!
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Teacher Edition

Are you ready for Chapter 1?
Take the Chapter 1 Pretest to find out.

What You Know

Entry–Level Assessment Assess students’ prior
knowledge at the beginning of a chapter
with one of the following options.

• Preview

2 Preview

STEP

Initial Assessment Assess students’ knowledge
at the beginning of the year with the
Diagnostic and Placement Tests.

Student Study Guide

1 Chapter Pretest

5

14 - 8 =

6

10 - 4 =

7

9-6=

8

13 - 5 =

-1

-1

-1

-1

-1

-1

-1

-1

-1

-5 - 4 = -9

You know how to multiply.

Lesson 1-5

Example: 5 · 9 = 45

To multiply integers, multiply their
absolute values , then write the sign
of the product.

TRY IT!
9

6·6=

10

8·4=

11

10 · 4 =

12

7·5=

Find -3 · 4.
|-3| · |4| = 3 · 4
= 12
So, -3 · 4 = -12.

3

T10 Teacher Handbook

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Comprehensive Assessment System

2

Chapter

Progress Check 3

5

Formative

(Lessons 5-5 and 5-6)

Find the area of each figure.
1

2

2.75 m
20 m

3 ft

A=

A=

1.5 ft

Progress Monitoring Determine if students are progressing
adequately as you teach each lesson. Use the assessments to
differentiate lesson instruction and practice.

triangle

3

4

rectangle
= 12.5 m, w = 5 m

b = 10 cm, h = 12 cm
Find the volume of each figure.
5

6

7

10 cm

2 cm

4.5 ft

4 cm
8 cm

6.25 cm

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Student Study Guide
• Progress Check
• Reflect

Teacher Edition
Intervention Strategy
Are They Getting It?
See It, Do It, Say It, Write It
Ticket Out the Door

Data-Driven Decision Making

3 ft

triangular prism

8

2 ft

V=
9

V=

rectangular prism

10

rectangular prism

b = 4 in., h = 6 in.,
h2 = 7.5 in.

= 0.5 yd, w = 1.2 yd,
h = 3 yd

= 16 mm, w = 25 mm,
h = 14 mm

V=

V=

V=

Solve.
11

PAPER GOODS A full box of 50 tissues has a length of 8 inches and
a height of 4.5 inches. If the width of the box is 3 inches, what is the
volume of the box of tissue?

12

LANDSCAPING A landscaper is laying sod on a triangular piece of
ground with a base length of 24 meters and a height of 45 meters.
What is the area?

Progress Check 3

3

Summative

Chapter

6

Chapter 6 Test

The data in the line graph shows the average price of a
dozen eggs in January 2003 to 2009. Use this graph to
answer Questions 1 and 2.
1

Between which years was the greatest decrease?

2

What was the price of a dozen eggs in 2007?

Annual Average Egg
Prices per Dozen

Price ($)

Summative Evaluation Assess student success
in learning the concepts in each chapter.

201

$3.20
$2.80
$2.40
$2.00
$1.60
$1.20
$0.80
$0.40

$20
0
20 3
0
20 4
0
20 5
0
20 6
07
20
0
20 8
09

•
•
•
•
•

7 cm

V=

Year

The histogram shows the number of books in 20 library
branches. Use this histogram to answer Questions 3 and 4.
3

• Chapter Test

4

How many libaries have 20,000 or more books?

In which interval(s) do the least number of libraries fall?

9
8
7
6
5
4
3
2
1
0
5–9 10–14 15–19 20–24 25–29 30+

The data in the table shows the sports played by a sixth
grade class. Create a bar graph to display the data. Use
the graph to answer Questions 5–7.

• Data-Driven Decision Making
• Alternative Assessment
• Assessment Masters:
Chapter Test

Sport
softball
basketball

5

Number of
Students
13
24

swimming

15

soccer

27

gymnastics

9

football

11

What is the difference in the number of students who play the most
popular and the least popular sport?

6

About how many students play football?

7

How many students were surveyed?

262

Number of Books (thousands)

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Teacher Edition

Books at Local Libraries

Frequency

Student Study Guide

Chapter 6 Test

Comprehensive Assessment System T11

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Contents

Chapter

1

Number and Operations

1-1 Number Groups ................................................................................4
1-2 Compare and Order Numbers ........................................................9
Progress Check 1 .............................................................................14
1-3 Add Integers ....................................................................................15
1-4 Subtract Integers .............................................................................21
Progress Check 2 .............................................................................26
1-5 Multiply Integers ............................................................................27
1-6 Divide Integers ................................................................................31
Progress Check 3 .............................................................................35
1-7 Solve One-Step Equations .............................................................36
1-8 Graph Inequalities ..........................................................................41
Progress Check 4 .............................................................................45
Chapter Test .....................................................................................46

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Contents
Chapter

2

Decimals

2-1 Decimals and Place Value ..............................................................50
2-2 Compare and Order Decimals ......................................................55
Progress Check 1 .............................................................................60
2-3 Round Decimals ..............................................................................61
2-4 Add and Subtract Decimals ..........................................................65
Progress Check 2 .............................................................................69
2-5 Multiply Decimals ..........................................................................70
2-6 Divide Decimals ..............................................................................75
Progress Check 3 .............................................................................79
Chapter Test .....................................................................................80
Albuquerque, New Mexico

v

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Contents
Chapter

3

Fractions and Mixed Numbers

3-1 Greatest Common Factors .............................................................84
3-2 Simplify Fractions ...........................................................................89

Progress Check 1 .............................................................................94
3-3 Mixed Numbers ..............................................................................95
3-4 Compare and Order Fractions ......................................................99
Progress Check 2 ...........................................................................104
3-5 Multiply Fractions ........................................................................105
3-6 Divide Fractions ............................................................................109
Progress Check 3 ...........................................................................113
3-7 Least Common Multiples ............................................................114
3-8 Add Fractions ................................................................................119
3-9 Subtract Fractions .........................................................................125
Progress Check 4 ...........................................................................131
Chapter Test ...................................................................................132

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Niagara Falls, New York


Contents
Chapter

4

Real Numbers

4-1 Rational Numbers .........................................................................136
4-2 Fractions and Decimals ................................................................141
Progress Check 1 ...........................................................................146

4-3 Decimals and Percents .................................................................147
4-4 Fractions and Percents .................................................................151
Progress Check 2 ...........................................................................156
4-5 Squares and Square Roots ...........................................................157
4-6 Compare and Order Real Numbers ...........................................161
Progress Check 3 ...........................................................................165
Chapter Test ...................................................................................166
Charleston, South Carolina

vii

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Contents
Chapter

5

Measurement and Geometry

5-1 The Customary System ................................................................170
5-2 The Metric System ........................................................................175
Progress Check 1 ...........................................................................180
5-3 Convert Between Systems ...........................................................181
5-4 Perimeter ........................................................................................185
Progress Check 2 ...........................................................................190
5-5 Area .................................................................................................191
5-6 Volume............................................................................................195
Progress Check 3 ...........................................................................201

5-7 Surface Area ...................................................................................202
5-8 The Coordinate Plane ...................................................................209
Progress Check 4 ...........................................................................213
Chapter Test ...................................................................................214

viii

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St. Augustine, Florida


Contents
Chapter

6

Probability and Statistics

6-1 Line Graphs ...................................................................................218
6-2 Frequency Tables and Histograms .............................................225
Progress Check 1 ...........................................................................230
6-3 Bar Graphs .....................................................................................231
6-4 Circle Graphs .................................................................................237
Progress Check 2 ...........................................................................242
6-5 Mean, Median, and Mode ...........................................................243
6-6 Count Outcomes ...........................................................................249
6-7 Probability......................................................................................255
Progress Check 3 ...........................................................................261
Chapter Test ...................................................................................262

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1

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Table of Contents
Using the Assessment Masters ................................ Aii
Using the Diagnostic and Placement Test .....Aiii–A1
Diagnostic and Placement Test .........................A2–A9
Chapter 1 Pretest .........................................................A10–A12
Chapter 1 Chapter Test ...............................................A13–A15
Chapter 2 Pretest .........................................................A16–A18
Chapter 2 Chapter Test ...............................................A19–A21
Chapter 3 Pretest .........................................................A22–A24
Chapter 3 Chapter Test ...............................................A25–A27
Chapter 4 Pretest .........................................................A28–A30

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Chapter 4 Chapter Test ...............................................A31–A33
Chapter 5 Pretest .........................................................A34–A36
Chapter 5 Chapter Test ...............................................A37–A39
Chapter 6 Pretest .........................................................A40–A42
Chapter 6 Chapter Test ...............................................A43–A45

Ai

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Using the Assessment Masters
Math Triumphs Assessment Masters are an important tool in the evaluation of specific areas
of student need, appropriate placement, and determining student success and readiness
for transitioning back to on-level curriculum. The assessments in this book provide a
comprehensive system that allows for diagnostic, formative, and summative assessment.

Diagnostic and Placement Test
For each

part, mark

Student

the box und

Name

er the num

ber of cor

rectly ans

wered que

stions.

0 1 2

3 4 5
6 7 8

Geometry
Spatial Rea and
soning

Measure
, Inc.

ment

Companies

Probability

and Statisti

of The McG

raw-Hill

cs

© Glencoe/M
cGraw-Hill,

a division

Mark the

total num

ber correc

t below.

0 1 2
3 4 5
6 7 8
9 10 11
12 13 14
15 16 17
18 19 20
21 22 23
24 25 26
Key: Con
27 28 29
30
sider this

Total

student for…

Math Triu
Algebra 1
Glencoe

Math Trium

phs

mphs: Fou
nda

Strategic

Algebra

1

tions for
Algebra
1
Interventio
n—See pag
e

Aix for ma

terials list

Diagnostic

.

and Placem

ent Tests

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PDF Pass
12/9/08

NAME ____

Chapter

________

1

Chapter

________

________

________

Test

________

________

________

3:46:44
PM

DATE ____
____

________

5

Multiplica

tive Identit

ty of Ad

dition

6

y

Associativ
e Proper
ty
of Multip
lication

n.

8

anies, Inc.
McGraw-H
ill Comp

11 -26
+ (-17)
=

Find eac
h produc
t or quotien
t.

10 -3
+ 10 =
12 27 (-8

13 -7
· 10 =

on of The
Copyright
© Glencoe/M
cGraw-Hill,
a divisi

5(6 + 2)

Find eac
h sum or
difference
.
9 -22
- 16 =

15 50 ÷
(-5) =

Solve eac
h equatio
n.

17 n =
-5 · (-7
), n =
19 c =

36 ÷ 12,

c=

21 w =
–6 + 8, w
=
23 d =
67 - 29
+ 7, d =

A10_A15_HS
MT_GeC01_
890860.indd

Using the Assessment Masters

tive Proper

Apply the
Distributiv
e Proper
ty to sim
7 7(4 +
plify eac
3)
h expres
sio

Math Trium
phs

An Answer Key at the end of this book provides reductions of

each student page with answers.

4

Commuta

____

)=

14 -12
·8=
16 -54
÷

(-6) =

18 p =
-4

0 · 2, p =

-12_
20 t = ___
,
6

22 z =
-45
24 h =

-3

t=

÷ 9, z =

42 + 19 -

131, h =

Chapter
13

1 Test

A13

11/22/08

9:16:23
AM

Math Triumphs

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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

The Chapter Pretests and Tests cover concepts and skills
presented in each chapter of Math Triumphs. The Chapter

Pretest is multiple-choice format and can be used to determine
specific student needs and assign specific chapters and lessons
within that chapter. The Chapter Test is short-answer format
and can serve as a summative assessment of the contents of each
chapter to determine student success in learning that chapter’s
concepts and skills.

________

Chapter 1

Verify the
expressio
ns are equ
1 (44 +
al. Name
23) + 16
the proper
= 44 + (23
ty shown
+ 16)
.
2 43 ·
0=0
Write an
equation
to show
eac
3 Distrib
h proper

ty.
utive Pro
perty

Chapter Pretest and Test

Aii

Algebra 1

Patterns
, Relation
ship
and Alge
braic Thin s,
king

See page Aiii for further explanation and instructions
for administering and scoring this test.

Answer Key

Placement
Test

Number,
Ope
Quantitativ ration, and
e Reason
ing

Copyright

The Diagnostic and Placement Test contains 30
mutiple-choice questions and addresses the content
strands: Number, Operations, and Quantitive Reasoning;
Patterns, Relationships, and Algebraic Thinking;
Geometry and Spatial Reasoning; Measurement; and
Probability and Statistics.

Scoring
Guide

nt Test

The Diagnostic and Placement Test assesses concepts
and skills that are presented in an on-level curriculum.
To determine if a student has successfully mastered
prerequisite concepts and skills for Algebra 1, use the
Algebra 1 Diagnostic and Placement Test. The results
can be used to determine whether students require
intensive intervention, and, if so, what specific content
strands students need.


Using the Diagnostic and Placement Test
Algebra 1
Placement Test

The Diagnostic and Placement Test provides tools to assist teachers in making placement

decisions with regard to one of the following on-level, strategic intervention, or intensive
intervention resources.

Math Connects
A balanced basal program that provides on-level and strategic intervention instruction.

Math Triumphs
An intensive intervention program for students two or more years behind
grade level.
This assessment provides valuable diagnostic information that teachers may also find helpful
throughout the school year. The Learning Objectives found before the test provide further
information on using these tests as diagnostic tools. These tests are also available in Spanish at
glencoe.com.

Placement Decisions

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

In making placement decisions for a student, consider a variety of evidence, such as the student’s
mathematics grades, classroom observations, teacher recommendations, portfolios of student
work, standardized test scores, and placement test scores. Use the results of these placement
tests in conjunction with other assessments to determine which mathematics course best fits a
student’s abilities and needs.
These tests can help determine whether or not students need intervention as well as the level of
intervention required. Through strategic intervention, teachers can work with students using
on-level content but strategically choose which content strand(s) need further development.
Sometimes a student may struggle with a particular strand, but overall the student is able to
perform on-level.
Intensive intervention is used with students who are struggling with most or all strands of math
content and are unable to work on grade-level. These students will need alternative intervention

materials to help meet their needs. These materials, such as the Math Triumphs program, offer
alternatives that will accelerate achievement in mathematics.

Test Content
These placement tests measure ability, but they are not achievement tests. They cover prerequisite
concepts, not every concept found in Glencoe mathematics textbooks or in your state standards.
Mathematics concepts are introduced, developed, and reinforced in consecutive courses. These
placement tests measure student mastery of concepts and skills that have been introduced or
developed in the student’s previous mathematics courses and that are further developed in the
next course.

Math Triumphs

Using the Diagnostic and Placement Test

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Aiii


When to Use the Diagnostic and Placement Test
In most situations, these placement tests are given near the end of the current course in
order to help determine student placement for the following year. You can also use these
tests in special situations, such as a student transferring into your school mid-year or
entering middle school with advanced mathematics ability.

Interpreting Scores
When interpreting scores on the placement tests,
consider the student’s score on each part, as well as the
total score. Scoring Guide Masters before each test can

be reproduced and used to record each student’s score.
A sample of a completed Scoring Guide is shown to the
right.

Placement
Test

Scoring
Guide

Student Name

For each part, mark the box under the number of correctly answered questions.

Sample Score and Placement Analysis
On a sample test, suppose a student scored 8 out of 8
questions correct in Number, Operations, and Quantitative
Reasoning; 5 in Patterns, Relationships, and Algebraic
Thinking; 2 in Geometry and Spatial Reasoning; 3 in
Measurement; and 2 in Probability and Statistics. The total
number correct was 20 out of 30. Note: There is not a direct
correlation between the score for each strand and the total
test score. Use the total score for class placement decisions
and the score by strand when working on particular
objectives.

Key: Consider this student for…

Math Triumphs: Foundations for Algebra 1
Algebra 1 Strategic Intervention—See page Aix for materials list.

Glencoe Algebra 1
Math Triumphs

Diagnostic and Placement Tests

Av

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Sample Analysis
This student scored well in the first two parts of the test, but lower in the remaining
sections. If these results are similar to other assessments, this student is likely to need
intervention materials and may find the grade-level material too challenging without any
intervention. The teacher should also note that the student scored particularly low in the
Geometry and Spatial Reasoning. This student may require intensive intervention when
these skills are taught.

Using Placement Tests for Diagnostic Purposes
These placement tests also provide valuable diagnostic information for classroom
teachers. Reproducible learning-objective charts list the learning objective for each
test question and can be found before each test. By marking each question the student
answered incorrectly, you can see which objectives the student has not mastered.

Aiv

Using the Diagnostic and Placement Test

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Math Triumphs

Algebra 1
Placement Test

Placement
Test
Scoring

Student Name

Algebra 1

Guide

For each part, mark the box under the number of correctly answered questions.
0 1 2 3 4 5 6 7 8
Number, Operation, and
Quantitative Reasoning

Patterns, Relationships,
and Algebraic Thinking

Geometry and
Spatial Reasoning

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Measurement

Probability and Statistics

Mark the total number correct below.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Total
Key: Consider this student for…

Math Triumphs: Foundations for Algebra 1
Algebra 1 Strategic Intervention—See page Aix for materials list.
Glencoe Algebra 1
Math Triumphs

Diagnostic and Placement Tests

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Av


Placement
Test
Learning
Objectives

Strand

Student Name

Algebra 1

Question
Number

3

Objective
Compare and order rational numbers in various forms
including integers, percents, and positive and negative
fractions and decimals.
Approximate (mentally and with calculators) the value of
irrational numbers
as they arise from problems situations
__
(such as π, √2 ).
Express numbers in scientific notation, including negative
exponents, in appropriate problems situations.

4

Express numbers in scientific notation, including negative
exponents, in appropriate problems situations.

5

Select appropriate operations to solve problems involving
rational numbers and justify the selections.

6

Select appropriate operations to solve problems involving
rational numbers and justify the selections.

7

Use appropriate operations to solve problems involving
rational numbers in problem situations.

8

Use multiplication by a constant factor (unit rate) to represent
proportional relationships.

1

2

Number,
Operation, and
Quantitative
Reasoning

10

Geometry
and Spatial
Reasoning

Avi

11

Compare and contrast proportional and non-proportional linear
relationships.
Translate among verbal, tabular, graphical, and algebraic
representations of functions, and describe how such aspects
of a function as slope and y-intercept appear in different
representations.
Estimate and find solutions to application problems involving
percents and other proportional relationships such as
similarity and rates.

12

Solve systems of two linear equations in two variables and
relate the systems to pairs of lines that intersect, are parallel,
or are the same, in the plane.

13

Find and evaluate an algebraic expression to determine
any term in an arithmetic sequence (with a constant rate of
change.

14

Generate similar figures using dilations including
enlargements and reductions.

15

Graph dilations, reflections, and translations on a coordinate
plane.

16

Use facts about the angles created when a transversal cuts
parallel lines to explain why the sum of the measures of the
angles in a triangle is 180 degrees, and they apply this fact
about triangles to find unknown measures of angles.

17

Use geometric concepts and properties to solve problems in
fields such as art and architecture.

Diagnostic and Placement Tests

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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

9

Patterns,
Relationships,
and Algebraic
Thinking

In the column on the left, mark the questions that the
student answered incorrectly.


Geometry
and Spatial
Reasoning

Question
Number
18

Objective
Use geometric concepts and properties to solve problems in
fields such as art and architecture.

19

Locate and name points on a coordinate plane using ordered
pairs of rational numbers.

20

21
Measurement

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Probability and

Statistics

22

Algebra 1
Placement Test

Strand

Find lateral and total surface area of prisms, pyramids, and
cylinders using concrete models and nets (two-dimensional
models).
Estimate measurements and use formulas to solve application problems involving lateral and total surface area and
volume.
Use the Pythagorean Theorem to solve real-life problems.

23

Use proportional relationships in similar two-dimensional
figures or similar three-dimensional figures to find missing
measurements.

24

Describe the resulting effects on perimeter and area when
dimensions of a shape are changed proportionally.

25

Find the probability of dependent and independent events.

26

Use theoretical probabilities and experimental results to
make predictions and decisions.

27

Select the appropriate measure of central tendency or range
to describe a set of data and justify the choice for a particular
situation.

28

Draw conclusions and make predictions analyzing trends in
scatterplots.

29

Select and use an appropriate representation for presenting
and displaying relationships among collected data including
line plots, line graphs, stem and leaf plots, histograms, and
Venn diagrams, with and without the use of technology.

30

Evaluate methods of sampling to determine validity of an
inference made from a set of data.

Math Triumphs

Diagnostic and Placement Tests

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Avii


0LACEMENT
4EST
Algebra 1

$IAGNOSTIC
#HART
Student
Performance
Level
Intensive
Intervention

Number of
Questions
Correct
0–17

Strategic
18–23
Intervention

24 or more

Use Math Triumphs: Foundations for Algebra 1 to
accelerate the achievement of students who are below
grade level. Students should follow a personalized
remediation plan. A variety of materials and instructional
methods are recommended. For example, instruction
and practice should be provided in print, technology,
and hands-on lessons.
Use the additional Intervention and Remediation
materials listed on the next page. This list of materials
can provide helpful resources for students who struggle
in the traditional mathematics program. Strategic
intervention allows students to continue to remain in
the Glencoe Algebra 1 program, while receiving the
differentiated instruction they need. Teaching Tips and
other resources are also listed in the Teacher Edition.
Use Glencoe Algebra 1. This student does not require
overall intervention. However, based on the student’s
performance on the different sections, intervention
may be required. For example, a student who missed
3 or more questions in the Measurement section may
require extra assistance as you cover these skills
throughout the year.

A Special Note About Intervention
When using diagnostic tests, teachers should always question the reason behind
the students’ scores. Students can struggle with mathematics concepts for a variety
of reasons. Personalized instruction is recommended for English language learners,
students with specific learning disabilities, students with certain medical conditions, or
for those who struggle with traditional instructional practice. Teachers should always

consider the needs of the individual student when determining the best approach for
instruction and program placement.

Aviii

Diagnostic and Placement Tests

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Algebra 1

Suggestions for Intervention and Remediation