x + 2 =a3
+ 4b× 4 =c16y
b = 6x + 3
Table of Contents
Teacher Resources
Solving Systems of Equations . . . . . . . . . . . . . . . . . . . . 93
Research on the Effectiveness of Intervention . . . . . .2
Creating Equations from Number Patterns
(Functions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Developing Students’ Mathematics Vocabulary . . . .5
Vocabulary Warm-up Activities . . . . . . . . . . . . . . .6
Using Concrete Models to Introduce
Mathematical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 10
Types of Manipulatives and How They
Are Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Differentiating Student Guided Practice . . . . . . . . . 14
Response to Intervention in the
Mathematics Classroom. . . . . . . . . . . . . . . . . . . . . 14
Differentiation by Specific Needs . . . . . . . . . . . . 14
Solving Real-Life Mathematical Problems . . . . . . . . 16
Problem-Solving Steps . . . . . . . . . . . . . . . . . . . . . . 16
Problem-Solving Strategies . . . . . . . . . . . . . . . . . . 17
Playing Games in Mathematics . . . . . . . . . . . . . . . . . . 19
Games Used in This Program . . . . . . . . . . . . . . . . 20
Program Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Lesson Organization . . . . . . . . . . . . . . . . . . . . . . . . 22
Planning for Intervention . . . . . . . . . . . . . . . . . . . . 28
Pacing Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Correlation to Mathematics Standards . . . . . . . . . . 33
How to Find Your State Correlations . . . . . . . . . 33
NCTM Standards Correlation Chart . . . . . . . . . 33
Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Diagnostic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Lesson Plans
Translating Descriptions into Expressions . . . . . . . 105
Translating Descriptions into Equations. . . . . . . . . 111
Graphing Linear Equations . . . . . . . . . . . . . . . . . . . . . 118
Interpreting Linear Equation Graphs . . . . . . . . . . . . 123
Generating Multiple Representations of
Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Finding the Slope of a Line . . . . . . . . . . . . . . . . . . . . . 135
Graphing a Line in Slope-Intercept Form. . . . . . . . 141
Finding x- and y-Intercepts . . . . . . . . . . . . . . . . . . . . . 148
Working with Transformations . . . . . . . . . . . . . . . . . 153
Using Similar Figures to Solve Problems. . . . . . . . . 160
Determining Angle Measures When
Parallel Lines Are Cut by a Transversal . . . . . . . . . . 165
Determining Missing Angles in Triangles
and Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Applying the Pythagorean Theorem . . . . . . . . . . . . 178
Solving Literal Equations for Perimeter, Area,
Surface Area, and Volume . . . . . . . . . . . . . . . . . . . . . . 183
Solving Problems with Mixed Units
(Conversions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
Determining Range and Central Tendency . . . . . . 195
Generating and Interpreting
Box-and-Whiskers Plots . . . . . . . . . . . . . . . . . . . . . . . . 201
Working with Exponents and Scientific Notation . . . 45
Generating and Interpreting Scatter Plots
and Lines of Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
Identifying, Comparing, and Ordering
Rational and Irrational Numbers . . . . . . . . . . . . . . . . 51
Interpreting and Predicting from Graphs . . . . . . . 213
Adding and Subtracting Rational Numbers
in Word Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Multiplying and Dividing Rational Numbers
in Word Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Probability with Independent and
Dependent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Appendices
Appendix A: References Cited . . . . . . . . . . . . . . . . . . 226
Working with Real-Life Application of Percents . . 70
Appendix B: Answer Key . . . . . . . . . . . . . . . . . . . . . . . 227
Simplifying Algebraic Expressions . . . . . . . . . . . . . . . 75
Appendix C: Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Solving Single and Multistep Linear Equations . . . 81
Appendix D: Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Solving and Graphing Linear Inequalities . . . . . . . . 88
Appendix E: Contents of Teacher Resource CD . . . 246
© Teacher Created Materials
#11134 (i2645)—Targeted Mathematics Intervention, Level 8
i
x
TEACHER RESOURCES
6
= 1y
4y
b = 6x + 3
Research on the Effectiveness
of Intervention
For many people, “Math is right up there with snakes, public speaking, and heights” (Burns 1998).
Mention the word mathematics and the room grows silent, people squirm in their seats, and small
beads of sweat appear on their temples. In an effort to keep others from the disappointment that will
certainly ensue, a disclaimer is spoken, “I am not good at math.” Parents hire tutors because they do
not feel comfortable helping their fifth graders with math homework. People hail the mathematics
professor as a god-like figure. These behaviors certainly can discourage both learners and teachers
alike. Even so, mathematics cannot be ignored because it is relevant every day in real life. In fact,
it might be nearly impossible to live a day without math. Counting spare coins in a jar, estimating
the cost of an item to include tax, and converting recipes for larger groups are just a few of the ways
mathematics enters people’s daily routines. Because of this, it is imperative that this mathematics
phobia be put to rest. But how can these misconceptions about mathematics be overcome? The
cure for mathematics phobia might be as simple as providing students with the necessary skills and
opportunities to be successful in mathematics.
The Need for Intervention
The National Council for Teachers of Mathematics (NCTM) has some high expectations or goals
for students. They want students to become mathematical problem solvers, learn to communicate
and reason mathematically, and make connections (NCTM 2000). The task of teaching mathematics
in classrooms today appears more difficult because today’s educators understand that curriculum
must be differentiated to meet the needs of all students. Students come into mathematics class with
different levels of readiness, learning styles, and interests. To meet the needs of all learners, teachers
must provide varied levels of time, structure, support, and complexity with different intensities for
these learners (Coleman 2003). Targeted Mathematics Intervention incorporates differentiated
activities throughout the lessons. The differentiation is seen in the focus on vocabulary, small-group
guided practice, and open-ended activities.
The learning differences among students can be illustrated in several ways. The basic mathematics
facts require some memorization. Some students have not been taught memorization skills. English
language learners have language and communication issues, and this affects every content area,
including math. Students who have processing difficulties find the mathematics symbols and numerals
confusing. Other students perform poorly in mathematics simply because they have low self-esteem
and short attention spans. Some students have even developed the attitude that mathematics is scary
or boring based on previous experiences. Personality dictates that some students are passive learners
or are simply disorganized. This makes the task of teaching mathematics complicated for teachers.
The one-size-fits-all curriculum and instruction cannot possibly accomplish the goal of reaching
all students (Tomlinson 2003). For these reasons, mathematics intervention is necessary. Still, the
question remains: What kinds of mathematics interventions work?
2
#11134 (i2645)—Targeted Mathematics Intervention, Level 8
© Teacher Created Materials
x
TEACHER RESOURCES
6
= 1y
4y
b = 6x + 3
Differentiating Student Guided Practice
Response to Intervention in the Mathematics Classroom
All students learn differently and struggle with different mathematical concepts. In one classroom,
teachers can have students who are above grade level, on grade level, below grade level, and English
language learners. Not all below-grade-level students struggle in the same areas and fit into the same
“category.” Because of this, many of the same researchers who created the Reading First initiative
developed a system of identification known as Response to Intervention (RTI). The RTI model
supports the idea that teachers should look for curricular intervention designed to bring a child
back up to speed as soon as he or she begins having difficulties. “RTI has the potential then to allow
disabilities to be identified and defined based on the response a child has to the interventions that
are tried” (Cruey 2006). Depending on the levels of difficulty they are having with the mathematics
curriculum, students are classified as Tier 1, 2, or 3. Specific definitions of these tiers differ from state
to state, but the following are general descriptions.
Tier 1 students are generally making good progress toward the standards, but may be experiencing
temporary or minor difficulties. These students may struggle only in a few of the overall areas of
mathematical concepts. They usually benefit from peer work and parental involvement. They would
also benefit from confidence boosters when they are succeeding. Although they are moving ahead,
any problems that do arise should be diagnosed and addressed quickly in order to ensure that these
students continue to succeed and do not fall behind.
Tier 2 students may be one or two standard deviations below the mean on standardized tests. These
students are struggling in various areas and these struggles are affecting their overall success in a
mathematics classroom. These students can usually respond to in-class differentiation strategies and
do not often need the help of student study teams.
Tier 3 students are seriously at risk of failing to meet the standards as indicated by their extremely
and chronically low performance on one or more measures of the standardized test. These students
are often the ones who are being analyzed by some type of in-house student assistance team in order
to look for overall interventions and solutions. In the classroom, these would be the students who are
having difficulties in most of the assignments and failing most of the assessments.
Differentiation by Specific Needs
Below-Grade-Level Students
Below-grade-level students will probably need concepts to be made more concrete for them. They
may also need extra work with manipulatives and application games. By giving them extra support
and understanding, these students will feel more secure and have greater success.
• Allow partner work for oral rehearsal of solutions and allocate extra time for guided practice.
• Allow for kinesthetic activities where they organize the step-by-step processes on flash cards
before they actually use the information to solve problems.
• Have easy-to-follow notes of the most important procedural information already made up for
these students to add to.
14
#11134 (i2645)—Targeted Mathematics Intervention, Level 8
© Teacher Created Materials
x + 2 =a3
+ 4b× 4 =c1
+3
TEACHER RESOURCES
Program Overview (cont.)
Pacing Plans (cont.)
Once a pacing plan is created based on known needs of the students or the results of the Diagnostic
Test, teachers are able to focus on the lessons that correlate with the items for which students did
not demonstrate mastery. The Diagnostic Test 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 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 skills may come easier to the majority of the students in the classroom. If this is the
case, allow less time for the practice and application of those skills and choose instead to move
ahead to the next lesson in the program.
• Skip those lessons or concepts for which students have demonstrated mastery on the
Diagnostic Test.
• Reduce the number of problems that students practice in the Student Guided Practice Book.
• Spend more time with the problem-solving activities, learning games, and vocabulary
activities.
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. This would allow students to learn from each other while
reinforcing their understanding of the concepts.
• Use the PowerPoint slide shows as additional lessons for reteaching the skills or concepts.
• Correct with students the problems in the Student Guided Practice Book or have students
model the problems on the board or overhead.
• Review all the standardized test preparation with students and have students resolve the
problems that were incorrect.
© Teacher Created Materials
#11134 (i2645)—Targeted Mathematics Intervention, Level 8
29
x + 2 =a3
+ 4b× 4 =c16y
b = 6x + 3
Diagnostic Test
The following is a small version of the Diagnostic Test. A full-size version of this test is provided in the
Student Guided Practice Book from pages 6–12.
●
1
2
●
A
6.4 × 106
B
0.64 × 10–5
C
6.4 × 10–6
D
0.64 × 105
2
3
F
G
,
13
3
5
●
,3
2
3
2
3
,
12 costumes
11 costumes
10 costumes
13 costumes
Solve.
45.5% of 105 =
, –2, –8, 1.000
–2, –8, 1.000, 3
13
3
Mindy purchased 16 14 yards
of fabric. If she needs 1 12
yards to make a costume,
how many costumes can
be made?
F
G
H
J
A
47.775
B
4,777.5
C
477.75
D
4.7775
13
3
, 1.000, –2, –8
13
3
H
–8, –2, 1.000,
J
–8, –2, 1.000, 3
2
3
,3
,
2
3
13
3
Together Susan and Lisa
purchased 10 23 yards of
fabric. If Lisa purchased
6 15 yards of fabric, how many
yards did Susan purchase?
A
B
C
D
© Teacher Created Materials
4
Order the numbers from least
to greatest.
3
3
●
●
What is 0.0000064 in
scientific notation?
6
●
Simplify.
–4(5x – 6y) – (4x – 9y)
F
–24x + 33y
16 38 yards
G
–24x – 33y
7
4 15
yards
H
–24x + 15y
J
–24x – 15y
4 48
yards
16 13
15 yards
#11134 (i2645)—Targeted Mathematics Intervention, Level 8
39
4y
4 ×3
3x4 = 1y6
x+2=3
b = 6x + 3
Diagnostic Test (cont.)
7
●
8
●
10
●
Solve.
4x
5
+ 3 = –1
Which equation can be used
to calculate the 20th term?
6, 9, 12, 15, . . .
A
5
F
x+3
B
–3 15
G
3x + 3
C
35
H
3x
D
–5
J
2x + 6
1
Which graph represents the
given inequality?
11
●
Which describes the
expression?
4x – 10
3x + 2 > 8
F
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
G
H
J
9
●
40
Mrs. Petty has a rectangular
garden with an area of 24 ft.2
and a perimeter of 20 ft.
What are the dimensions of
the garden?
12
●
A
4 times the number x
less than 10
B
4 times the number x,
minus 10
C
4 divided by the
number x, minus 10
D
4 divided by the
number x less than 10
Which equation describes
the statement?
7 more than a number x
is equal to the difference
between 25 and a number x
A
12 ft. × 2 ft.
F
7x = 25 – x
B
3 ft. × 8 ft.
G
7 + x = x – 25
C
10 ft. × 2 ft.
H
7 – x = 25 + x
D
6 ft. × 4 ft.
J
x + 7 = 25 – x
#11134 (i2645)—Targeted Mathematics Intervention, Level 8
© Teacher Created Materials
b6 × 8 =348 +
a 5+=b8 c 6y ÷
16
2+6=
LESSON 1
x+2=3
4 × 4 = 16
b = 6x + 3
Working with Exponents and
Scientific Notation
Materials
• Student Guided
Practice Book
- Exploring Exponents and
Scientific Notation
(pages 13–14;
page013.pdf )
- Exponents and Scientific
Notation (page 15;
page015.pdf )
- Working Backwards
(page 16; page016.pdf )
- Standardized Test
Preparation 1 (page 17;
page017.pdf )
- Match It! Directions
(page 143; page143.pdf )
• Punchouts folder
Learning Objectives
• Write and evaluate numbers in exponent form.
• Write whole numbers or decimals in scientific notation.
Warm-up Activity
Skill: Measurement
1. Settle the students for the day. Then, show the students a
box with a rectangular base. Ask how to find the volume
of the box.
2. Draw a box on the board or overhead projector. Point out
that it is a rectangular prism. Write the formula “volume
of rectangular prism = length × width × height” on the
board or overhead.
3. Draw this table below the prism on the board or overhead.
Length
Width
Height
Volume
3 in.
4 in.
3 in.
x
x
2 cm
3 cm
48 cm3
5 ft.
x
5 ft.
75 ft.3
5 mm
7 mm
x
105 mm3
- Match It! Cards
(matchit.pdf )
- Counters (counters.pdf )
• Transparency folder
- Working Backwards
(trans01.pdf )
• PowerPoint folder
on the CD
- Working with Exponents
and Scientific Notation
(lesson01.ppt) (optional)
• Game board
- Top Hits
• four spinners
• paper and pencils
15 min.
4. Place student in pairs and have them find the unknown
variables.
5. Ask for a student volunteer to solve for each of the
unknown values.
Vocabulary
10 min.
Complete the Chart and Match (page 6) vocabulary activity
using the words below. Definitions of these words are included
on pages 243–245 in this book and in the Student Guided
Practice Book.
• base
• scientific notation
• exponent
• standard form
• power
© Teacher Created Materials
#11134 (i2645)—Targeted Mathematics Intervention, Level 8
45
16
4 =+
3x 6y ÷43y ==4 ×2mx
83
6×
2+6=
LESSON 1
Working with Exponents and
Scientific Notation (cont.)
Whole-Class Skills Lesson
25 min.
Use the directions below or the PowerPoint presentation to teach this lesson.
1. Tell students, “Today, you will be using exponents to write numbers in scientific notation.”
2. Define the terms base, exponent, and power by writing the problem, 2 • 2 • 2 = 23. Point out that
in 23 = 8, 2 is the base, 3 is the exponent, and 8 is the power of 2. The exponent shows how many
times the base needs to be multiplied by itself.
3. Have students turn to Exploring Exponents and Scientific Notation (SGPB pages 13–14) and work
problems 1–4. Review the problems with the students.
4. Ask the students to look at problems 5–7. As you discuss the example row, explain that the first
term must be greater than or equal to 1 but less than 10. Ask students to work with partners to
complete the next three rows. Then, ask volunteers to explain how they filled in those rows.
5. Discuss how to solve problem 8 without having to take the middle steps. Ask students, “Where
is the decimal point in 70,000? Which direction do you move the decimal point when writing the
whole number in scientific notation? How many places does the decimal point move?” Discuss
the answers with the students. When discussing number 8, use problems 5–7 to show how to
write a number in scientific notation without having to take the middle steps.
6. Have students complete problems 9 and 10. Then, ask volunteers to explain the solutions.
7. Write this pattern on the board or overhead.
•
•
•
•
23 = 8
22 = 4
21 = 2
20 = 1
• 2–1 = 12
• 2–2 = 12 • 12 = 14
• 2–3 = 12 • 12 • 12 = 18
8. Ask students what is happening to the exponents as you move from 23 to 2–3. (The exponent is
decreasing by 1.) Ask students what is happening to the powers as you move from 23 to 2–3.
(The powers are being divided by 2.)
9. In pairs, have students solve problems 11–14. Then, ask volunteers to explain how to solve the
problems.
10. Review with the students how to write the number 75,000 in scientific notation. (7.5 • 104) Ask
the students to look at problems 15–17. As you discuss the two example rows, explain that
the numerator must be greater than or equal to 1 but less than 10. Ask students to work with
partners to complete the next three rows. Then, ask volunteers to explain their work.
11. Discuss with the students how to solve problem 18 without having to take the middle steps.
When discussing 0.0000056 = 5.6 • 10-6, use the chart to show how to determine a way to write a
decimal in scientific notation without having to take the middle steps. This would be the time to
discuss moving the decimal to the right.
12. Have pairs complete problems 19 and 20. Then, discuss how to solve the problems.
46
#11134 (i2645)—Targeted Mathematics Intervention, Level 8
© Teacher Created Materials
b6 × 8 =348 +
a 5+=b8 c 6y ÷
16
2+6=
LESSON 1
Working with Exponents and
Scientific Notation (cont.)
Differentiated Guided Practice
20 min.
● Below Level—Teacher Directed
• Have students work in pairs on Exponents and Scientific Notation (SGPB page 15).
• While students are writing exponents, have them identify the bases and ask how many times
they are being multiplied. As students are writing whole numbers in scientific notation,
ask them to identify the direction in which each decimal point is moving. While students
are writing decimals in scientific notation, ask them to identify the direction in which each
decimal point is moving.
■ On/Above Level—Student Directed
• Have students work independently to complete Exponents and Scientific Notation
(SGPB page 15).
4 × 4 = 16
• If students finish early, have them draw another prism and write at least five questions about it.
Then, have students switch papers and answer each other’s questions.
Problem Solving
• Place the Problem-Solving Strategy
Transparency: Working Backwards on the
overhead. Use the guiding callouts on page
48 to introduce the strategy to the students.
A copy of this transparency is also included
in the Student Guided Practice Book on
page 16. Students can follow along and
make notes as you review the transparency.
Test Preparation
x+2=3
20 min.
10 min.
• Have students complete Standardized
Test Preparation 1 in the Student Guided
Practice Book on page 17. Give them
about seven minutes. Then, have students
trade papers and grade their work.
© Teacher Created Materials
Learning Game
20 min.
Match It!
• While students are completing the test
preparation questions, set up four game
stations. Each station needs one Top Hits
game board, a set of Match It! Cards, and a
handful of Counters.
• Review with students the Match It!
Directions (SGPB page 143). Answer any
questions that students have about how to
play. You may want to model one round of
play.
• Allow them time to play the game.
Move among the students, checking for
understanding, as they complete the
mathematics problems. Make sure you
stop the students with about five minutes
left so that they can clean up the game
stations.
#11134 (i2645)—Targeted Mathematics Intervention, Level 8
47
16
4 =+
3x 6y ÷43y ==4 ×2mx
83
6×
2+6=
LESSON 1
Working with Exponents and
Scientific Notation (cont.)
Problem-Solving Transparency Callouts
❶
❷
Read and discuss
the example given in
the top paragraph.
Together, see if
you can figure out
everyone’s ages. Show
them how to write
algebraic expressions
and equations to
determine the
answers.
Read and discuss the information across the top of
the page, reinforcing how working backwards can be
a useful tool for solving problems. Ask students if any
of them have ever worked a maze backwards. Why do
they think that it is easier to work mazes backwards?
y 4y
3xx
Use with Lesson 1.
Working Backwards
The Problem
Read the problem
to determine the
information already
known. Suggest
that underlining
or highlighting the
information from the
problem may be useful.
Help students decide
which information should
be underlined. Check to
make sure that what the
students underlined is
the same as what is listed
in the Understanding the
Problem section.
48
Read and discuss the
chart to see how it
was used to solve the
problem. Discuss with
students how working
backwards helped them
locate the solution to the
problem.
Working backwards can help you solve problems that have a lot of events or several steps
where some information is missing. Many times, this information is missing at the beginning
of the problem. To solve these problems, you can usually start with the answer and work your
way backwards to fill in the missing information. This strategy is very helpful in dealing with
a sequence of events or when each piece of information is related to the one before it. For
example, Paul is four years older than Antonio, but Kaya is two years younger than Antonio. If
Kaya is eight, how old is Paul? In this case, everyone’s age is related to everyone else’s age.
Working backwards will help you solve this problem.
Problem: Library Books
❸
❹
Addie started reading a library book on
Monday. On Sunday, she has 15 pages
left to read to complete the book. Use the
information to calculate the number of pages
in the book. On Monday, she read 20 pages.
On Tuesday, she read ½ the number of pages
she did on Monday. On Wednesday, she
read 3 fewer pages than she has to read
on Sunday. On Thursday, she read 4 more
pages than she read on Tuesday. On Friday,
she read the sum of the number of pages
she read on Tuesday and Wednesday. On
Saturday, she read half the sum of the
number of pages she read on Thursday and
Friday. How many pages are in the book?
Understanding the Problem
On Monday, she read 20 pages.
15 + 20 = 35
On Tuesday, she read 12 the
number of pages she did on
Monday. ( 12 of 20 = 10)
35 + 10 = 45
On Wednesday, she read 3
fewer pages than she has to
read on Sunday. (15 – 3 = 12)
45 + 12 = 57
On Thursday, she read 4
more pages than she read on
Tuesday. (10 + 4 = 14)
57 + 14 = 71
On Friday, she read the sum of
the number of pages she read
on Tuesday and Wednesday.
(10 + 12 = 22)
71 + 22 = 93
On Saturday, she read half the
sum of the number of pages she
read on Thursday and Friday.
( 12 of 14 + 22 = 18)
93 + 18 = 111
Her book has 111 pages.
• What do we know?
Addie has 15 pages left to read.
Information is given on how many
pages are read each day.
Reflecting and Generalizing
By starting with the number of pages she has
left, we were able to work backwards to find
the answer. This strategy can be used when
we know the end result but don’t know the
starting point. You can check your answer by
working forward through the problem to see if
you reach the correct number.
• What do we need to find out?
How many pages are in the book?
Planning and Communicating a Solution
Begin with the information you know, the
number of pages she has left to read to
complete the book. Start with the 15 pages
she has left to read.
❺
Reinforce with students
the importance of
reflecting on how the
solution was reached.
Talk about the best types
of problems for using this
strategy.
Extension
Discuss why you need to start calculating the
number of pages read on Monday instead of
Saturday.
#11134 (i2646)—Targeted Mathematics Intervention, Transparencies
© Teacher Created Materials
❻
#11134 (i2645)—Targeted Mathematics Intervention, Level 8
Have students answer the extension question in pairs. Then,
discuss the question as a whole class.
© Teacher Created Materials
b6 × 8 =348 +
a 5+=b8 c 6y ÷
16
2+6=
LESSON 1
Working with Exponents and
Scientific Notation (cont.)
Student Pages and Punchouts Needed for the Lesson
x+2=3
4 × 4 = 16
x+2=3
b = 6x + 3
Name ______________________________________
4 × 4 = 16
Exploring Exponents and Scientific Notation
2. 14 • 14 • 14 = ____________________________________________________________
2. 11 • 11 • 11 ______________________________________________________________
3. 3 • 3 • 3 • 3 • 3 • 3 • 3 = ______________________________________________________
3. 7 • 7 • 7 • 7 _______________________________________________________________
4. 9 • 9 • 9 • 9 = _____________________________________________________________
4. 2 • 2 • 2 • 2 • 2 • 2 • 2 _______________________________________________________
Directions: Solve.
Problem: Library Books
5. 8–2 = ___________________________________________________________________
Directions: Complete the chart to write each number in scientific notation.
example: 16,000
Show
Multiplication
1.6 • 10,000
Show Multiplying by 10s
Scientific Notation
1.6 • 10 • 10 • 10 • 10
1.6 • 10
7. 930,000,000 = ___________________________________________________________
5. 50,000
8. 0.0000005 = _____________________________________________________________
6. 130,000
9. 10,200,000 = ____________________________________________________________
7. 25,000,000
10. 0.00000102 = ____________________________________________________________
Directions: Use the rectangular prism to
answer questions 11–14. Remember that
the volume of a rectangular prism is
length x width x height.
Directions: Write each number in scientific notation.
8. 70,000 = ________________________________________________________________
1
2
Addie started reading a library book on
Monday. On Sunday, she has 15 pages
left to read to complete the book. Use the
information to calculate the number of pages
in the book. On Monday, she read 20 pages.
On Tuesday, she read ½ the number of pages
she did on Monday. On Wednesday, she
read 3 fewer pages than she has to read
on Sunday. On Thursday, she read 4 more
pages than she read on Tuesday. On Friday,
she read the sum of the number of pages
she read on Tuesday and Wednesday. On
Saturday, she read half the sum of the
number of pages she read on Thursday and
Friday. How many pages are in the book?
Directions: Write each in scientific notation.
4
10 units
10 units
15 units
9. 104,000 = _______________________________________________________________
11. Find the volume of the rectangular prism, and write it in scientific notation.
10. 335,000,000 = ___________________________________________________________
11. 3–2 = ___________________________________________________________________
• What do we know?
Addie has 15 pages left to read.
Information is given on how many
pages are read each day.
_______________________________________________________________________
12. 4–3 = ___________________________________________________________________
13. Multiply all the measurements of the original rectangular prism by 100. Find the volume
of the new rectangular prism, and write it in scientific notation.
13. 5–2 = ___________________________________________________________________
• What do we need to find out?
How many pages are in the book?
_______________________________________________________________________
14. 2–4 = ___________________________________________________________________
Planning and Communicating a Solution
14. Multiply all the measurements of the original rectangular prism by 1,000. Find the volume
of the new rectangular prism, and write it in scientific notation.
Begin with the information you know, the
number of pages she has left to read to
complete the book. Start with the 15 pages
she has left to read.
_______________________________________________________________________
© Teacher Created Materials
#11334—Targeted Mathematics Intervention, Guided Practice Book
13
© Teacher Created Materials
Exploring Exponents
and Scientific
Notation
(SGPB pages 13–14;
page013.pdf )
x+2=3
#11334—Targeted Mathematics Intervention, Guided Practice Book
15
16
On Wednesday, she read 3
fewer pages than she has to
read on Sunday. (15 – 3 = 12)
45 + 12 = 57
On Thursday, she read 4
more pages than she read on
Tuesday. (10 + 4 = 14)
57 + 14 = 71
On Friday, she read the sum of
the number of pages she read
on Tuesday and Wednesday.
(10 + 12 = 22)
71 + 22 = 93
On Saturday, she read half the
sum of the number of pages
she read on Thursday and
Friday. ( 12 of 14 + 22 = 18)
93 + 18 = 111
Reflecting and Generalizing
By starting with the number of pages she has
left, we were able to work backwards to find
the answer. This strategy can be used when
we know the end result but don’t know the
starting point. You can check your answer
by working forward through the problem to
see if you reach the correct number.
Extension
2
●
3
●
If a circle is cut into 6 equal pieces,
what would be the degree measure of
one of the center angles?
A
60°
B
30°
C
15°
D
6°
#11334—Targeted Mathematics Intervention, Guided Practice Book
4
●
In a preseason hockey tournament,
Quentin got 5 goals in a total of
12 shots. If he continues to score
at this rate for the rest of the season,
how many goals should he have after
180 shots?
F
G
H
J
Solve for x.
–2x = –3x + 5x – 3
F
3
x= 4
G
x=–2
5
H
x=–3
5
J
x=1
What is 6,340,000 expressed in
scientific notation?
5
●
6
●
60 goals
65 goals
70 goals
75 goals
Leticia and Renée have part-time
jobs. Renée earns $60 more each
week than Leticia. In 10 weeks,
Leticia earns as much as Renée in
4 weeks. Which system of equations
will determine the weekly earnings
of Leticia, l, and Renée, r?
A
l + r = 60
10l = 4r
C
l + 10 = r + 4
l = r + 60
B
10l = 4r
l = r + 60
D
10l = 4r
r = l + 60
Pick one question from this test.
Explain how and why you chose
your answer.
A
63.4 x 10–5
B
6.34 x 105
________________________
C
6.34 x 106
________________________
D
6.34 x 10–6
________________________
________________________
________________________
Discuss why you need to start calculating
the number of pages read on Monday
instead of Saturday.
© Teacher Created Materials
Working Backwards
(SGPB page 16;
page016.pdf )
Exponents and
Scientific Notation
(SGPB page 15;
page015.pdf )
4 × 4 = 16
On Tuesday, she read the
number of pages she did on
Monday. ( 12 of 20 = 10)
35 + 10 = 45
Her book has 111 pages.
Understanding the Problem
_______________________________________________________________________
12. Double all the measurements of the original rectangular prism. Find the volume of the
new rectangular prism, and write it in scientific notation.
Directions: Solve.
1
●
On Monday, she read 20 pages. 15 + 20 = 35
The Problem
6. 2–6 = ___________________________________________________________________
Whole Number
b = 6x + 3
Standardized Test Preparation 1
Working Backwards
1. 8 • 8 • 8 • 8 • 8 = ___________________________________________________________
4 × 4 = 16
x+2=3
Name ______________________________________
Working backwards can help you solve problems that have a lot of events or several steps
where some information is missing. Many times, this information is missing at the beginning
of the problem. To solve these problems, you can usually start with the answer and work your
way backwards to fill in the missing information. This strategy is very helpful in dealing with
a sequence of events or when each piece of information is related to the one before it. For
example, Paul is four years older than Antonio, but Kaya is two years younger than Antonio. If
Kaya is eight, how old is Paul? In this case, everyone’s age is related to everyone else’s age.
Working backwards will help you solve this problem.
Directions: Write each in exponent form.
1. 4 • 4 • 4 • 4 • 4 ____________________________________________________________
b = 6x + 3
Name ______________________________________
Exponents and Scientific Notation
Directions: Write each expression in exponent form.
4 × 4 = 16
x+2=3
b = 6x + 3
Name ______________________________________
________________________
© Teacher Created Materials
#11334—Targeted Mathematics Intervention, Guided Practice Book
17
Standardized Test
Preparation 1
(SGPB page 17;
page017.pdf )
b = 6x + 3
Counters
Match It! Cards
.BUDI*U2VFTUJPO
.BUDI*U2VFTUJPO
)BMDVUTBQJ[[BJOUP
FRVBMQJFDFT8IBUJTUIF
EFHSFFNFBTVSFNFOUPG
POFPGUIFDFOUFSBOHMFT
*OUIFGJSTUIBMGPGIJT
CBTLFUCBMMHBNF+VBO
NBLFTCBTLFUTPVUPG
TIPUT*GIFDPOUJOVFT
UPTDPSFBUUIJTSBUFGPS
UIFSFTUPGUIFHBNFIPX
NBOZCBTLFUTXJMMIFNBLF
JGIFTIPPUTUJNFT Match It! Directions
What You Need
• Top Hits game board
• spinner (Divide the spinner into 4 parts and write the numbers 1–4 on it.)
• Match It! cards
• Counters or something else to use as game markers
.BUDI*U2VFTUJPO
-JTUUIFGPMMPXJOH
TFUPGGSBDUJPOTBOE
EFDJNBMTJOPSEFSGSPN
HSFBUFTUUPTNBMMFTU
mm • pencils and paper
Object of the Game
• Match math problems and their answers. Use your memory to do this. Be the first player to move
around the board.
.BUDI*U2VFTUJPO
.BUDI*U2VFTUJPO
.BUDI*U2VFTUJPO
Setting Up the Game
• Place the game board in the middle of all the players.
• Shuffle the 36 cards and place them facedown. Arrange them in 6 rows with 6 cards in each row.
• Each player places a game marker on START.
How to Play the Game
4PMWFGPS[
4PMWFGPSZ
4JNQMJGZUIFFYQSFTTJPO
m[m[[m
ZmZ
mNm
N
• The youngest player goes first. Then, play passes to the left.
• For each turn, flip over two cards. Make sure that everyone can see the cards when you flip them.
• If you flip a mathematics problem, solve it. Then, try to find the correct answer. If you flip
an answer, remember where it is and try to find the correct problem. You are trying to match
problems with their answers.
.BUDI*U2VFTUJPO
• If you do not have a match, your turn is over.
.BUDI*U2VFTUJPO
.BUDI*U2VFTUJPO
• If you make a pair, spin the spinner and move that many spaces.
• When you stop, record the number of weeks. This is the number of weeks that your record is
at #1!
-BTUXFFLPVUPG
FWFSZTUVEFOUTQBTTFE
UIFHFPNFUSZUFTUJO
.S-JOTDMBTT*G
TUVEFOUTUPPLUIFUFTU
IPXNBOZQBTTFE • Your turn is over after you move. Keep any pairs you find.
&YQSFTTJO
TDJFOUJGJDOPUBUJPO
How to Win the Game
• The first player to land on END wins!
• Second place goes to whomever has the most matches. (If the winner had the most matches, there
is no second place!)
.JUBTGBUIFSXFJHIT
QPVOET5IJTJT
QPVOETMFTTUIBOUISFF
UJNFTIFSXFJHIU)PX
NVDIEPFT.JUBXFJHI • Third place goes to whomever has albums with the most weeks at #1.
© Teacher Created Materials
#11334—Targeted Mathematics Intervention, Guided Practice Book
143
Match It! Directions
(SGPB page 143;
page143.pdf )
© Teacher Created Materials
#11134 (i2649)—Targeted Mathematics Intervention, Punchouts
© Teacher Created Materials
Match It! Cards
(matchit.pdf )
#11134 (i2649)—Targeted Mathematics Intervention, Punchouts
© Teacher Created Materials
Counters
(counters.pdf )
#11134 (i2645)—Targeted Mathematics Intervention, Level 8
49
16
4 =+
3x 6y ÷43y ==4 ×2mx
83
6×
2+6=
LESSON 1
Working with Exponents and
Scientific Notation (cont.)
PowerPoint Presentation Slides
Warm-Up Activity
Today’s Lesson
How do you find the
volume of the box?
Volume of the rectangular prism
= length x width x height
V = lwh
We will warm up today by
working with volume.
Working with Exponents
and Scientific Notation
height
width
length
Length
2 in.
Width
Height
5 in.
4 in.
Volume
Length
x
V = lwh
x=2
•
5•
Width
Height
Volume
in.3
2 in.
5 in.
4 in.
x
3 cm
4 cm 60 cm3
40
How can you find the missing variable?
4
V = l •w •h
60 = x • 3 • 4
60 = 12x
x = 5 in.
x = 40 in.3
Length
Width
Height
Volume
2 in.
5 in.
4 in.
x
x
3 cm
4 cm 60 cm3
7 ft.
x
3 ft.
How can you find the missing variable?
V = l•w•h
84 = 7• x • 3
84 = 21x
4 ft. = x
Width
Height
9 mm
2 mm
Volume
x
x
14 in.
6 in.
252 in.3
2 ft.
8 ft.
x
144 ft.3
What is a base?
3∙3∙3∙3 = 34
Today you will be working
with exponents to write
numbers in scientific notation.
Length
7 mm
84 ft.3
What is another way to write 3∙3∙3∙3?
Whole-Class Skills Lesson
Find the unknown variables.
exponent
power
34 = 81
34 = 81
The base is the factor that you multiply.
What is an exponent?
The exponent tells how many times to
multiply the base.
base
What is a power?
The result of multiplying the base.
How do you write these expressions in
exponent form?
3∙3∙3∙3∙3∙3
36
Write each number in scientific notation.
Whole
Show
Number Multiplication
71,000
5∙5∙5∙5
52∙52∙52
54
523
7.1 x 10,000
Show Multiplying
by 10
Scientific
Notation
7.1 ∙ 10 ∙ 10 ∙ 10 ∙ 10
7.1 ∙ 104
30,000
The first term must be
greater than or equal to
1,150,000
1 but also less than 10.
Scientific Notation
Scientific Notation
3,500,000
900,000
3.5 x 106
Where is the decimal point?
The first term must be
greater than or equal to
1 but also less than 10.
The exponent tells how
many places to move
the decimal.
Which direction do you move the decimal point when
writing the whole number in scientific notation?
How many places does the decimal point move?
9.0 x 105
Write the numbers below in scientific
notation.
5,600
125,000
7,500,000
34,000
5.6 x 103
1.25 x 105
What pattern do you see?
33 = 27
32 = 9
31 = 3
30 = 1
30 = 1
3–1 =
3–1 =
3.4 x 104
3–2 =
3–2 =
3–3 =
3–3 =
0.0032
3.2 x 10–3
0.056
5.6 x 10–2
0.0000098
9.8 x 10–6
0.00071
7.1 x 10–4
Write 94,000 in scientific
notation.
Remember the first number must be a
number greater than 1 but less than 10.
How many places do we need to move
the decimal?
9.4 x 104
#11134 (i2645)—Targeted Mathematics Intervention, Level 8
What happens to the exponents
as you move from 33 to 3–3?
The exponent is decreasing by 1.
31 = 3
7.5 x 106
Write the numbers below in scientific
notation.
50
33 = 27
32 = 9
Scientific Notation
0.0000042
Where is the decimal point?
What happens to the powers
as you move from 33 to 3–3?
Which direction do you move the decimal point when
writing the whole number in scientific notation?
The powers are being divided by 3.
How many places does the decimal point move?
4.2 x 10–6
Write the numbers below in scientific
notation.
How do you solve the problems?
0.00729
7.29 x 10–3
2,250
2.25 x 103
423,000
4.23 x 105
0.00063
6.3 x 10–4
© Teacher Created Materials
x+2=3
4 × 4 = 16
b = 6x + 3
Name ______________________________________
Exploring Exponents and Scientific Notation
Directions: Write each expression in exponent form.
1. 4 • 4 • 4 • 4 • 4 ____________________________________________________________
2. 11 • 11 • 11 ______________________________________________________________
3. 7 • 7 • 7 • 7 _______________________________________________________________
4. 2 • 2 • 2 • 2 • 2 • 2 • 2 _______________________________________________________
Directions: Complete the chart to write each number in scientific notation.
Whole Number
example: 16,000
Show
Multiplication
Show Multiplying by 10s
Scientific Notation
1.6 • 10,000
1.6 • 10 • 10 • 10 • 10
1.6 • 104
5. 50,000
6. 130,000
7. 25,000,000
Directions: Write each number in scientific notation.
8. 70,000 = ________________________________________________________________
9. 104,000 = _______________________________________________________________
10. 335,000,000 = ___________________________________________________________
Directions: Solve.
11. 3–2 = ___________________________________________________________________
12. 4–3 = ___________________________________________________________________
13. 5–2 = ___________________________________________________________________
14. 2–4 = ___________________________________________________________________
© Teacher Created Materials
#11334—Targeted Mathematics Intervention, Guided Practice Book
13
x+2=3
4 × 4 = 16
b = 6x + 3
Exploring Exponents and
Scientific Notation (cont.)
Directions: Complete the chart.
Show Division
Show Division by
Tens
Scientific Notation
example 0.0008
8
10,000
8
10 • 10 • 10 • 10
8 • 10–4
example 0.034
3.4
100
3.4
10 • 10
3.4 • 10–2
Decimal
15. 0.000006
16. 0.00057
17. 0.000000044
Directions: Write each in scientific notation.
18. 0.0000056 = _____________________________________________________________
19. 0.00103 = _______________________________________________________________
20. 0.00009 = _______________________________________________________________
14
#11334—Targeted Mathematics Intervention, Guided Practice Book
© Teacher Created Materials
x+2=3
4 × 4 = 16
b = 6x + 3
Name ______________________________________
Exponents and Scientific Notation
Directions: Write each in exponent form.
1. 8 • 8 • 8 • 8 • 8 = ___________________________________________________________
2. 14 • 14 • 14 = ____________________________________________________________
3. 3 • 3 • 3 • 3 • 3 • 3 • 3 = ______________________________________________________
4. 9 • 9 • 9 • 9 = _____________________________________________________________
Directions: Solve.
5. 8–2 = ___________________________________________________________________
6. 2–6 = ___________________________________________________________________
Directions: Write each in scientific notation.
7. 930,000,000 = ___________________________________________________________
8. 0.0000005 = _____________________________________________________________
9. 10,200,000 = ____________________________________________________________
10. 0.00000102 = ____________________________________________________________
Directions: Use the rectangular prism to
answer questions 11–14. Remember that
the volume of a rectangular prism is
length x width x height.
10 units
10 units
15 units
11. Find the volume of the rectangular prism, and write it in scientific notation.
_______________________________________________________________________
12. Double all the measurements of the original rectangular prism. Find the volume of the
new rectangular prism, and write it in scientific notation.
_______________________________________________________________________
13. Multiply all the measurements of the original rectangular prism by 100. Find the volume
of the new rectangular prism, and write it in scientific notation.
_______________________________________________________________________
14. Multiply all the measurements of the original rectangular prism by 1,000. Find the volume
of the new rectangular prism, and write it in scientific notation.
_______________________________________________________________________
© Teacher Created Materials
#11334—Targeted Mathematics Intervention, Guided Practice Book
15
x+2=3
4 × 4 = 16
b = 6x + 3
Name ______________________________________
Working Backwards
Working backwards can help you solve problems that have a lot of events or several steps
where some information is missing. Many times, this information is missing at the beginning
of the problem. To solve these problems, you can usually start with the answer and work your
way backwards to fill in the missing information. This strategy is very helpful in dealing with
a sequence of events or when each piece of information is related to the one before it. For
example, Paul is four years older than Antonio, but Kaya is two years younger than Antonio. If
Kaya is eight, how old is Paul? In this case, everyone’s age is related to everyone else’s age.
Working backwards will help you solve this problem.
Problem: Library Books
On Monday, she read 20 pages. 15 + 20 = 35
The Problem
Addie started reading a library book on
Monday. On Sunday, she has 15 pages
left to read to complete the book. Use the
information to calculate the number of pages
in the book. On Monday, she read 20 pages.
On Tuesday, she read ½ the number of pages
she did on Monday. On Wednesday, she
read 3 fewer pages than she has to read
on Sunday. On Thursday, she read 4 more
pages than she read on Tuesday. On Friday,
she read the sum of the number of pages
she read on Tuesday and Wednesday. On
Saturday, she read half the sum of the
number of pages she read on Thursday and
Friday. How many pages are in the book?
Understanding the Problem
• What do we know?
Addie has 15 pages left to read.
Information is given on how many
pages are read each day.
• What do we need to find out?
How many pages are in the book?
Planning and Communicating a Solution
Begin with the information you know, the
number of pages she has left to read to
complete the book. Start with the 15 pages
she has left to read.
16
On Tuesday, she read 12 the
number of pages she did on
Monday. ( 12 of 20 = 10)
35 + 10 = 45
On Wednesday, she read 3
fewer pages than she has to
read on Sunday. (15 – 3 = 12)
45 + 12 = 57
On Thursday, she read 4
more pages than she read on
Tuesday. (10 + 4 = 14)
57 + 14 = 71
On Friday, she read the sum of
the number of pages she read
on Tuesday and Wednesday.
(10 + 12 = 22)
71 + 22 = 93
On Saturday, she read half the
sum of the number of pages
she read on Thursday and
Friday. ( 12 of 14 + 22 = 18)
93 + 18 = 111
Her book has 111 pages.
Reflecting and Generalizing
By starting with the number of pages she has
left, we were able to work backwards to find
the answer. This strategy can be used when
we know the end result but don’t know the
starting point. You can check your answer
by working forward through the problem to
see if you reach the correct number.
Extension
Discuss why you need to start calculating
the number of pages read on Monday
instead of Saturday.
#11334—Targeted Mathematics Intervention, Guided Practice Book
© Teacher Created Materials
4 × 4 = 16
x+2=3
b = 6x + 3
Name ______________________________________
Standardized Test Preparation 1
1
●
If a circle is cut into 6 equal pieces,
what would be the degree measure of
one of the center angles?
A
60°
B
30°
C
D
2
●
3
●
In a preseason hockey tournament,
Quentin got 5 goals in a total of
12 shots. If he continues to score
at this rate for the rest of the season,
how many goals should he have after
180 shots?
F
G
H
J
15°
6°
Solve for x.
–2x = –3x + 5x – 3
F
3
x= 4
G
x=–2
5
H
x=–3
5
J
x=1
What is 6,340,000 expressed in
scientific notation?
A
4
●
5
●
6
●
60 goals
65 goals
70 goals
75 goals
Leticia and Renée have part-time
jobs. Renée earns $60 more each
week than Leticia. In 10 weeks,
Leticia earns as much as Renée in
4 weeks. Which system of equations
will determine the weekly earnings
of Leticia, l, and Renée, r?
A
l + r = 60
10l = 4r
C
l + 10 = r + 4
l = r + 60
B
10l = 4r
l = r + 60
D
10l = 4r
r = l + 60
Pick one question from this test.
Explain how and why you chose
your answer.
63.4 x 10–5
________________________
5
B
6.34 x 10
________________________
C
6.34 x 106
________________________
D
6.34 x 10–6
________________________
________________________
________________________
© Teacher Created Materials
#11334—Targeted Mathematics Intervention, Guided Practice Book
17