Download Whole Numbers

7th Grade Unit 1 Resource
Grade Level: 7th Grade
Subject Area: 7th Grade Math
Lesson Title: Unit 1 Number and
Lesson Length: 9 days
Operations
THE TEACHING PROCESS
Lesson Overview
This unit bundles student expectations that address sets and subsets of rational
numbers, operations with rational numbers, and personal financial literacy
standards regarding sales tax, income tax, financial assets and liabilities records,
and net worth statements. According to the Texas Education Agency,
mathematical process standards including application, a problem-solving model,
tools and techniques, communication, representations, relationships, and
justifications should be integrated (when applicable) with content knowledge and
skills so that students are prepared to use mathematics in everyday life, society,
and the workplace.
During this unit, students use a visual representation to organize and display the
relationship of the sets and subsets of rational numbers, which include counting
(natural) numbers, whole numbers, integers, and rational numbers. Students also
apply and extend operations with rational numbers to include negative fractions
and decimals. Grade 7 students are expected to fluently add, subtract, multiply,
and divide various forms of positive and negative rational numbers which include
integers, decimals, fractions, and percents converted to equivalent decimals or
fractions for multiplying or dividing. Students also create and organize a financial
assets and liabilities record, construct a net worth statement, calculate sales tax
for a given purchase, and calculate income tax for earned wages.
Unit Objectives:
 Students will extend previous knowledge of sets and subsets using a visual
representation to describe relationships between sets of rational numbers.
 Students will add, subtract, multiply, and divide rational numbers fluently.
 Students will apply and extend previous understandings of operations to solve
problems using addition, subtraction, multiplication, and division of rational
numbers.
 Students will calculate the sales tax for a given purchase and calculate income
tax
for earned wages.
 Students will create and organize a financial assets and liabilities record and
construct a net worth statement.
Standards addressed:
TEKS:
 7.1A Apply mathematics to problems arising in everyday life, society, and the
workplace.
 7.1B Use a problem-solving model that incorporates analyzing given
information, formulating a plan or strategy, determining a solution, justifying the
solution, and evaluating the problem-solving process and the reasonableness
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of the solution.
7.1C Select tools, including real objects, manipulatives, paper and pencil, and
technology as appropriate, and techniques, including mental math, estimation,
and number sense as appropriate, to solve problems.
7.1D Communicate mathematical ideas, reasoning, and their implications using
multiple representations, including symbols, diagrams, graphs, and language
as appropriate.
7.1E Create and use representations to organize, record, and communicate
mathematical ideas.
7.1F Analyze mathematical relationships to connect and communicate
mathematical ideas.
7.1G Display, explain, and justify mathematical ideas and arguments using
precise mathematical language in written or oral communication.
7.2A Extend previous knowledge of sets and subsets using a visual
representation to describe relationships between sets of rational numbers.
7.3A Add, subtract, multiply, and divide rational numbers fluently
7.3B Apply and extend previous understandings of operations to solve problems
using addition, subtraction, multiplication, and division of rational numbers.
7.13A Calculate the sales tax for a given purchase and calculate income tax
for earned wages
7.13C Create and organize a financial assets and liabilities record and
construct a net worth statement
ELPS:
The student will
ELPS.c.1A: use prior knowledge and experiences to understand meanings in
English
ELPS.c.1C: use strategic learning techniques such as concept mapping, drawing,
memorizing, comparing,
contrasting, and reviewing to acquire basic and grade-level vocabulary
ELPS.c.1D: speak using learning strategies such as requesting assistance,
employing non-verbal cues, and using
synonyms and circumlocution (conveying ideas by defining or describing when
exact English words are not known)
ELPS.c.1E: internalize new basic and academic language by using and reusing it
in meaningful ways in speaking
and writing activities that build concept and language attainment
ELPS.c.2D: monitor understanding of spoken language during classroom
instruction and interactions and seek
clarification as needed
ELPS.c.2E: use visual, contextual, and linguistic support to enhance and confirm
understanding of increasingly
complex and elaborated spoken language
ELPS.c.3D: speak using grade-level content area vocabulary in context to
internalize new English words and build
academic language proficiency
ELPS.c.3H: narrate, describe, and explain with increasing specificity and detail as
more English is acquired
ELPS.c.4D: use pre-reading supports such as graphic organizers, illustrations,
and pre-taught topic-related vocabulary
and other pre-reading activities to enhance comprehension of written text
ELPS.c.4F: use visual and contextual support and support from peers and
teachers to read grade-appropriate
content area text, enhance and confirm understanding, and develop vocabulary,
grasp of language structures, and
background knowledge needed to comprehend increasingly challenging language
ELPS.c.4H: read silently with increasing ease and comprehension for longer
periods
ELPS.c.5B: write using newly acquired basic vocabulary and content-based
grade-level vocabulary
ELPS.c.5F: write using a variety of grade-appropriate sentence lengths, patterns,
and connecting words to combine phrases, clauses, and sentences in increasingly
accurate ways as more English is acquired
ELPS.c.5G: narrate, describe, and explain with increasing specificity and detail to
fulfill content area writing needs as more English is acquired.
Misconceptions:
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Some students may think the sum of any two rational numbers is always
greater than the two addends.
Some students may think the difference of any two rational numbers is always
less than the greater.
Some students may think the product of any two rational numbers is always
greater than the factors.
Some students may think the quotient of any two rational numbers is always
less than the dividend.
Some students may think the value of a property or home is a liability, rather
than an asset, if there is an outstanding mortgage on the property or home.
Some students may think the sales tax is the total cost, rather than the amount
added to the price to determine the total cost.
Some students may think that a percent may not exceed 100%.
Some students may think that a percent may not be less than 1%.
Some students may multiply a decimal by 100 moving the decimal two places
to the right when trying to convert it to a percent rather than dividing by 100
and moving the decimal two places to the left.
Some students may think the value of 43% of 35 is the same value of 43% of
45 because the percents are the same rather than considering that the wholes
of 35 and 45 are different, so 43% of each quantity will be different.
Some students may attempt to perform computations with percents without
converting them to equivalent decimals or fractions for multiplying or dividing.
Vocabulary:
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Counting (natural) numbers – the set of positive numbers that begins at one
and increases by increments of one each time {1, 2, 3, ..., n}
Earned wages – the amount an individual earns over given period of time
Financial asset – an object or item of value that one owns
Financial liability – an unpaid or outstanding debt
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Fluency– efficient application of procedures with accuracy
Income tax – a percentage of money paid on the earned wages of an
individual or business for the federal and/or state governments as required by
law
Integers – the set of counting (natural numbers), their opposites, and zero {-n,
…, -3, -2, -1, 0, 1, 2, 3, ..., n}. The set of integers is denoted by the symbol Z.
Net worth – the total assets of an individual after their liabilities have been
settled
Positive rational numbers – the set of numbers that can be expressed as a
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fraction , where a and b are whole numbers and b ≠ 0, which includes the
subsets of whole numbers and counting (natural) numbers (e.g., 0,
2,
etc.)
Rational numbers – – the set of numbers that can be expressed as a
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, where a and b are integers and b≠ 0, which includes the subsets of
fraction
integers, whole numbers, and counting (natural) numbers (e.g., -3, 0, 2,
,
etc.). The set of rational numbers is denoted by the symbol Q.
Sales tax – a percentage of money collected by a store (retailer), in addition to
a good or service that was purchased, for the local government as required by
law
Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3,
..., n}
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INSTRUCTIONAL SEQUENCE
Phase: Engage
List of Materials:
Engage Day 1 cards (one set per student, cut and bagged)
Activity: Common Characteristics
Prior to the lesson, the teacher has the following numbers written on the board (in
a scattered manner). There is a page of cards with these numbers in the lesson
(Engage Day 1 cards).
24
2
1
2
13
, 0, 50%, 4, 0.75, −5, 1 5, 32 , −0.25, 120%, 100,
3
2
Ask the students to group the numbers that have common characteristics and to
be ready to justify how they grouped the numbers.
What’s the teacher doing?
What are the students doing?
Walking around, answering questions
and monitoring students’ participation.
Sorting and grouping the numbers,
discussing the attributes with a partner.
Questions to ask after students have grouped the numbers:
 Why did you group these numbers together? Why are other numbers not
included in that group?
 How would you classify those numbers? Fractions? Whole numbers? Integers?
Percents? Mixed numbers? Proper fractions? Improper fractions? Decimals?
Exponents? A neutral number? Non-positive? Non-negative?
 Do any of these numbers simplify to another number? If so, which ones?
 Is there more than one way to represent these numbers?
 Can you give me another number (from the list or one you create) that would
also be in the group? Why?
Phase: Explore
List of Materials:
Engage Day 1 cards
Explore 1 Grouping Mat
Activity: Representing Rational Numbers
The teachers provides students with the Explore 1 Grouping Mat and ask
students to place the numbers from the Engage and the board (includes numbers
students added) on the mat in their appropriate location.
What’s the teacher doing?
What are the student’s doing?
The teacher is monitoring as students
work and is asking questions:
The students are placing the numbers
onto the mat and asking the teacher
and a partner questions about the
placement.
How can you identify a natural or
counting number?
It is a number that I learned when I first
started counting on my fingers: 1, 2, 3,
4, …
How can you identify a whole number?
A whole number is 0, 1, 2, 3, 4, …
The set of whole numbers contains all
of the counting numbers plus the
number 0.
How can you identify an integer?
I think of an integer as the positive and
negative “whole” numbers and 0.
How do you identify a rational number?
It can be written as a fraction of
integers, with the denominator not equal
to 0.
How can a decimal be a rational
number?
A terminating or repeating decimal can
be written as a fraction of integers, so it
is a rational number.
Phase: Explain (Part I & II)
List of Materials:
Explore 1: Day 1 Grouping Mat ( one per student)
Notes on Rational Numbers (one per student)
Notes on Rational Numbers (one per student)
Activity Part I: Justify
The students present their mats and justify where they placed the numbers.
The teacher asks the students the following questions:
How can you identify a natural or counting number?
It is a number that I learned when I first started counting on my fingers: 1, 2, 3, 4,
…
How can you identify a whole number?
A whole number is 0, 1, 2, 3, 4, …
The set of whole numbers contains all of the counting numbers plus the number 0.
How can you identify an integer?
I think of an integer as the positive and negative “whole” numbers and 0.
How do you identify a rational number?
It can be written as a fraction of integers, with the denominator not equal to 0.
How can a decimal be a rational number?
A terminating or repeating decimal can be written as a fraction of integers, so it is
a rational number.
In which set does a percent belong?
A percent is a rational number because it can be written as a ratio (fraction) of the
percent to 100.
Which set of number contains all of the other numbers? Why?
The set of rational numbers contains all of the other numbers (natural, whole,
integers) and also contains numbers that can be written as fractions of integers
(with the denominator not equal to zero).
Which set is a subset of the set of whole numbers? Why?
The set of natural numbers is the subset of the set of whole numbers because all
of the natural numbers are contained in the whole numbers.
What does the Venn diagram tell us about how the sets are related to each other?
The Venn diagram
The Venn diagram shows us that the natural numbers are contained in the whole
numbers, the whole numbers are contained in the integers, and the integers are
contained in the rational numbers.
Activity Part II: Notes on Rational Numbers
Then the teacher provides each student with a copy of Notes on Rational
Numbers for students to add to their journals. The students will determine if the
statements are true or false.
Also, have the students create a diagram of the rational number set and its
subsets then the students can write an explanation in their journals of where they
placed three numbers in the mat.
What’s the teacher doing?
What are the students doing?
The teacher is monitoring the students
as they work.
The students glue the Notes on
Rational Numbers in their journals.
Have students make vocabulary
flashcards from index cards, with the
definition on one side and the
vocabulary word on the other.
Students will make will make
vocabulary flashcards for :
Counting numbers (natural numbers),
integers, whole numbers, rational
numbers, positive rational numbers, and
keep them for future use.
Phase: Evaluate
List of Materials:
Paper
Activity: Performance Assessment 1:
Analyze the situation(s) described below. Organize and record your work for each
of the following tasks. Using precise mathematical language, justify and explain
each mathematical process.
1) Number sets are interrelated
a) Create a visual representation to organize and display the relationship of the
sets and subsets of numbers:
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counting (natural) numbers
integers
rational numbers
whole numbers
b) Write a description describing the relationships between sets of rational
numbers.
What’s the teacher doing?
What are the students doing?
The teacher is monitoring the students
as they work, asking questions as
needed.
The students are completing the
Performance Assessment individually.
Questions to check or understanding:
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How are counting (natural) numbers,
whole numbers, integers, and
rational numbers related?
What types of visual representations
can be used to represent the
relationships between sets and
subsets of numbers?
How can a number belong to the
same set of numbers but not
necessarily the same subset of
numbers?
Phase: Explore
List of Materials:
Another Look at Adding and Subtracting Fractions
Activity: Another Look at Adding and Subtracting Fractions.
Students will add and subtract fractions using Another Look at Adding and
Subtracting Fractions.
What’s the teacher doing?
Monitoring students as they work and
assisting students as needed.
What are the students doing?
Working in groups of two on the
assignment.
How can you rename the fractions so
that they have the same denominator?
I can write equivalent fractions with the
least common denominator
Phase: Explain
List of Materials:
Another Look at Adding and Subtracting Fractions
Activity: Teacher and Student Justifications
Teacher clarifies steps to work the problems and students will explain how they
worked the problems.
What’s the teacher doing?
What are the students doing?
The teacher is clarifying the steps to
working the problems as needed.
The students are explaining how they
worked each problem.
Which words in problems 9-14 indicated
the operations of addition or
subtraction?
In problem 9, the words “how much
does she have left” indicate subtraction.
In problem 10, the words, “how much in
5 minutes,” indicates addition, because
the other amounts were for 2 and 3
minutes.
In problem 11, the operation is
subtraction because part of the distance
is already traveled.
In problem 13, the operation is addition
for perimeter (the sum of the side
lengths).
In problems 12 and 14 both addition
and subtraction are used.
Phase: Explore/Explain
List of Materials:
Mac’s Brownies
Activity: Mac's Brownies
Display Mac’s Brownies, keeping the directions covered, and only displaying
the problem. Instruct students to read the problem and use mental math to
quickly select the range of numbers they believe represents the solution.
Survey students and tally the ranges students have estimated, as this will be
used in a later activity. Facilitate a class discussion for students to justify their
range of numbers selection.
What is the teacher doing?
1.
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Display teacher resource: Mac’s
Brownies, keeping the directions
covered, and only displaying the
problem. Instruct students to read
the problem and use mental math to
quickly select the range of numbers
they believe represents the solution.
Survey students and tally the ranges
students have estimated, as this will
be used in a later activity. Facilitate
a class discussion for students to
justify their range of numbers
selection.
Ask:
What method did you use to
determine the estimate? Possible
answer: I chose the 24 – 32 range
because know 1 pan yields 8
servings and 1 serving is 1 brownie
and there are 3 1 pans with 8
2
What are the students doing?
Students to read the problem and use
mental math to quickly select the range
of numbers they believe represents the
solution.
Students justify their range of numbers
selection.
brownies in each pan. Since 3 1 is
2
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between 3 and 4, did 3 x 8 = 24 and
4 x 8 = 32; etc.
Why did you select this method to
give an estimate? . I chose to
multiply the 2 whole numbers 3 1 is
2
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between; etc.
Is there another method that may
be used to determine an
estimate? Answers may vary. I
chose 3 x 8 = 24 because 3 1 comes
2
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after 3. I know the answer will be
more than 24; etc.
How are you going to determine
how many brownies Mac has
made using pictorial models?
A whole square can represent 1
pan, which can be divided into 8
equal parts – each one-eighth
fraction piece that represents 1
brownie. Since Mac made 3 1
2
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pans of brownies, 3 whole
squares plus half a whole square
will need to be the model for the
brownies.
How is the mathematical
operation of multiplication being
demonstrated as you cover the
whole square with the model for
brownie pieces? The operation of
multiplication is being demonstrated
because repeatedly creating multiple
groups of 8 brownies until 3 1 pans
2
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have been filled with 8 brownies on
each whole pan.
What multiplication number
sentence can be written to record
the relationship among the
numbers in the problem and how
these numbers relate to the model
created with the fraction squares?
The multiplication number sentence
is 3 1 pans x 8 brownies per pan =
2
28 total brownies.
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What does the 3 1 represent in
2
this problem and the in
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multiplication number sentence
you wrote? The number of pans
(groups) of brownies Mac has
baked.
How are the 8 brownies used in
this problem and the in
multiplication number sentence
you wrote? The 8 represents the
number of brownies (size of each
group – the factor, also known as
the multiplicand) placed on each
pan.
You only have half a pan. What
does this mean? Only going to
create a partial group -
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1
2
pan, which
means you only have 4 brownies
from the half a pan because 1 pan
yields 8 brownies.
How many brownies has Mac
made? The total number of
brownies Mac baked using 3 1 pans
2
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that yield 8 brownies per pan is 28
brownies.
***Facilitate a class discussion,
using the previously tallied results
gathered from students about their
estimate of the range of numbers
(20 to 30 brownies, 24 to 30
brownies, 24 to 32 brownies) from
the displayed teacher resource:
Mac’s Brownies. Demonstrate how
to determine the fraction of student
responses for each range of
numbers.
Ask:
What fraction of the class chose
this estimated range? Answers
may vary.
How did you choose your range
of numbers? Answers may vary. I
chose the estimated range of 24 –
32 brownies: knew 1 pan yielded 8
brownies so, I did 3 x 8 = 24 and 4 x
8 = 32 because 3 1 is between 3
2
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and 4; etc.
What fraction of the class did not
choose this estimated range?
Answers may vary.
What is the sum of the fraction of
Students demonstrate how to determine
the fraction of student responses for
each range of numbers.
Students discuss a class discussion
about the meaning and purpose of the
multiplication number sentence to make
a connection to the diagram. Students
record their actions on the fraction
diagrams.
students who did choose the
estimated range of 24 to 30
brownies and the fraction of
students who did not choose the
estimated range of 24 to 30
brownies? (The sum of the 2
fractions, the chosen estimated
range and the not chosen estimated
range should be 1 or very close to
1.)
2. Display the multiplication number
sentence 3 1 x 8 = 28 for the class
2
to see.
3. Using the displayed teacher
resource: Mac’s Brownies, facilitate
a class discussion about student
solutions to the problem. Without
using the formal algorithm, sketch a
picture to model the problem.
Facilitate a class discussion about
the meaning and purpose of the
multiplication number sentence to
make a connection to the diagram.
Ask:
 What is the multiplication number
sentence as a whole
communicating? (The number
sentence: 3 1 x 8 = 28 represents 3
2
1
2
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pans (groups) of 8 brownies in
each pan (number of objects in each
group) which produces 28 total
brownies.)
How is the addition number
sentence 8 + 8 + 8 + 4 = 24 related
to the multiplication number
sentence 3 1 x 8 = 28? (Both
2
number sentences are representing
3 1 groups of 8. The addition
2
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number sentence shows the number
in each group (8) being repeatedly
added. The multiplication number
sentence is communicating how
many groups of 8 are needed.)
What did you do in this problem?
(Combined multiple groups of a
given number.)
What is the mathematical term for
combining multiple groups of a
given number? (multiplication)
4. Explain to students that there are
various terms used in relation to
multiplication; the process of
combining multiple groups (factor,
also known as the multiplier) of a
given number (factor, also known as
the multiplicand) to produce a
product (the total quantity after
multiplying the factors).
Ask:
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What do the 3 1 pans represent in
2
the multiplication number
sentence: 3 1 x 8 = 28? (The
2
number of groups – first factor multiplier.)
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Explain to students that in
mathematics this is called a
“factor."
What is the purpose of this
factor? (This first factor, also
referred to as the multiplier,
indicates the number of groups.)
 What do the 8 brownies in the
multiplication number
sentence: 3 1 x 8 = 28
2
represent? (The number of
brownies that 1 pan yields - the
number in each group – the
second factor.)
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Explain to students that in
mathematics this is also called a
“factor."
What is the purpose of this
factor? (This factor, also referred
to as the multiplicand, indicates
the number or size of each
group.)
What do the 28 brownies in the
multiplication number
sentence: 3 1 x 8 = 28
2
represent? (The total number of
brownies there are from 3 1 pans
2
that yield 8 brownies per pan.)
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Explain to students that in
mathematics this is called a
“product."
What is the purpose of the
product? (It communicates the
total quantity after multiplying the
2 factors.)
For the multiplication number
sentence 2 1 x 6 = 14, what
3
word problem can be written
that would fit this number
sentence? Answers may vary.
Mac baked some of his famous
“Choco-Loco-Mint” brownies, but
decided to cut larger size
brownies so that 1 pan yields 6
servings. (Note: 1 brownie
represents 1 serving.) If Mac
baked 2 1 pans of “Choco-Loco3
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Mint” brownies, how many
brownies has he made?; etc.
How would you draw a
diagram that communicates
the meaning of the
multiplication number
sentence and shows the
relationship between the
number sentence and the
numbers in the problem? (The
diagram would be 3 whole
squares to represent the pans
with 1 whole square divided into
3 equal parts to show
1
3
pan.
Each pan would show 6 equal
size sections drawn to represent
the 6 brownies/pan.)
Example of the diagram:
1 pan
1 pan
1
3
pan
6 brownies 6 brownies 2 brownies
21
3
x
6
=
14
1st factor x 2nd factor = product
multiplier multiplicand
# of pans brownies/pan
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How would you label each
factor and the product in the
diagram to identify what each
factor and the product
represents? (In the
multiplication number sentence 2
1
3
x 6 = 14, the 2 1 is the first
3
factor, multiplier, and represents
the number of pans (groups).
The second factor is 6,
multiplicand, and represents the
number of brownies per pan (size
of each group). The product 14
represents the total number of
brownies in the 2 1 pans with 6
3
brownies per pan, total quantity
after multiplying the 2 factors.)
Phase: Explore/Explain
List of Materials:
Dividing Trim
Activity: Dividing Trim
What’s the teacher doing?
What are the students doing?
The teacher is monitoring students as
they work, assisting as needed and
asks the following guiding questions:
Students are completing the Dividing
Trim activity, recording their actions on
the fraction diagrams.
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How many lengths of one-fourth
yard are in a length of threefourths yard? (There are 3 lengths
of one-fourth yard in a length of
three-fourths yard.)
What does the number line
represent in this problem?
(Number of lengths of yards.)
How can you show lengths of
one-fourth yard on this number
line? (Begin at 0 and circle lengths
of one-fourth.)
What on the number line shows
how many lengths of one-fourth
yard are in a length of three-
Students discover the pattern for
dividing fractions with like
denominators.
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fourths yard? (The number of
circled lengths of one-fourth yard.)
What represents the size of each
group in this problem? (The length
of one-fourth yard.)
What represents the beginning
quantity, the dividend, on the
number line? (The mark and label
for three-fourths yard.)
What division number sentence
can be written to represent the
model? (
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3
4
yd. ÷
1
4
yd. = 3 lengths)
What is the dividend in the
number sentence and what is its
purpose? (Three-fourths, it
indicates the beginning quantityamount of ribbon available to by
cut.)
What is the divisor in the number
sentence and what is its purpose?
(One-fourth, it indicates the size of
each group which will be
successively removed from the
dividend.)
What is the quotient in the
number sentence and what is its
purpose? (3, it indicates the number
of groups of a given size that were
successively removed from the
dividend.)
***Facilitate a class discussion to
encourage the discovery of the
pattern for dividing fractions with like
denominators.
Ask:
 What do you notice about the
denominators of the dividend
and divisor? (They are like
denominators.)
 What do you notice about the
numerators of the dividend
and the divisor? (If you divide
the numerators from the dividend
and the divisor, you get the
quotient.)
 What do you notice about the
value of the quotient? (It is not
always smaller than the dividend
and the divisor.)
 What rule can you write for the
pattern in the problems? (If the
fractions have common
denominators, then divide the
numerators of the dividend and
the divisor to calculate the
quotient.)

Phase: Explore/Explain
List of Materials:
Extending Integers
Activity: Extending Integers
Distribute Extending Integers to pairs of students. Ask student to work together to
complete the handout.
What’s the teacher doing?
What are the students doing?
The teacher is monitoring students as
they work.
The students are working in pairs to
complete the assignment.
How do you show the sum of the
integers on the number line?
To sum -2, -4, and 4, start at 0 and draw
a ray from 0 to -2. Then draw a ray 4
units to the left to represent -4. From the
end of that arrow draw a ray 4 units to
the right to represent 4.
Phase: Explore/Explain
List of Materials:
Basic Operations Notes (one per student)
Basic Operations Notes and Skill Drills Part I (one per student)
Basic Operations Notes and Skill Drills Part II (one per student)
Activity: Basic Operations Notes and Skill Drills Part I
Distribute Basic Operations Notes to each student
Talk through the notes briefly, discussing any processes that are particularly
difficult for your students.
Distribute Basic Operations Skill Drills Part I to each student.
Ask students to work independently to complete the handout.
What’s the teacher doing?
What are the students doing?
Monitor students as they work, assisting
as needed.
Use the facilitation questions below as
needed.
 What would you do to solve ( -3 +
2)? Subtract 2 from 3 and keep the
sign of the larger one; etc.
 How would you solve an addition
problem with two negative
numbers? Add the two numbers
together and keep the negative sign.
 What would you do to solve (-6) (-2)? Add the opposite of the second
number. So, in this problem it would
become (-6) + 2, which would then
require you to follow the addition rule
of subtracting 2 from 6 and keeping
the negative sign; etc.
 How would you solve a
subtraction problem with one
negative number? Add the
opposite of the second number and
follow the addition rules to solve.
 How is multiplication and division
different from addition and
subtraction of integers? In
multiplication and division, you will
multiply or divide the numbers as
they appear. If they are both
negative numbers, the answer is
positive. If they are both positive
numbers, the answer is still positive.
If one number is positive and the
other is negative, the answer is
negative.
 What is the process you use to
add fractions? Find a common
denominator and then add the
numerators. If the problem has
mixed numbers, add the whole
numbers as well.
 What is the process you use to
subtract fractions? I will find a
common denominator and then
subtract the numerators. If the
After working several problems,
students can check their work with a
calculator.
Students complete the handout.

problem has mixed numbers, I
subtract the whole numbers as well.
How do you solve the problem
5
1
3
 2 ?. You will get a common
4
8
denominator of 8, so the new mixed
2
8
3
8
number will be 5  2 . Since you
cannot subtract 2 - 3, you will have
to borrow from the 5, making it a 4
and add the whole
it
8
2
to the making
8
8
10
. Now, you subtract the
8
7
8
numerators 10 and 3, getting . The
whole numbers are now 4 and 2, so
7
8
your final answer is 2 ; etc.



What is the process you use to
multiply fractions? Multiply the
numerators by numerators and the
denominators by denominators. If
the factors are mixed numbers, you
will need to first change them to an
improper fraction.
What is the process you use to
divide fractions? Take the
reciprocal of the second fraction and
then follow the multiplication
process.
3
5
How would you solve (  )  1
Change 1
9
?
10
9
to an improper fraction
10
19
. Then, take the reciprocal by
10
10
flipping the fraction over . Then,
19
of
multiply 3 by 10 and 5 by 19 to get
30
, and keep the negative sign
95
because of the integer rules. Then,
simplify the fraction to 
6
; etc.
19
Activity: Basic Operations Skill Drills Part II
Distribute Basic Operations Skill Drills Part II to each student.
Ask students to work independently to complete the handout.
What’s the teacher doing?
What are the students doing?
Monitor students as they work, assisting
After working several problems,
as needed.
students can check their work with a
Use the facilitation questions below as
calculator.
needed.
 How do you solve 2.56 +
3.293? (Line up the decimals,
add the numbers and move the
decimal point straight down.)
 What is the process for
subtracting decimals? (Line up
the decimals, subtract the
numbers and move the decimal
point straight down.)
 What is the process for
multiplying decimals? (Perform
the multiplication as you would
for whole numbers. Then, count
the number of digits to the right
of the decimal point in each
factor. Place the decimal point in
your answer so that your answer
has the same number of digits
after the decimal point.)
 How do you solve 55.04 ÷ 3.2?
(First, you move the decimal out
of the divisor, and then move the
decimal point the same number
of places in the dividend. Since
we have to move it one place in
3.2, we will move it one place for
55.04. So, we will divide 550.4 by
32. You get 172, but need to
bring the decimal point straight
up, so that your answer is 17.2.)
List of Materials:
Fractions without Distractions (one per student, in color preferably)
Activity: Fractions without Distractions
From the following website,
http://store.lonestarlearning.com/everything-else/conference-handouts/
select “Fractions” and print the second page “Fractions without Distractions and
provide one per student.”
Ask students to fold the notes into a little book and glue the notes into their journal.
What is the teacher doing?
What are the students doing?
The teacher is monitoring students as
Students are reviewing the process for
they work.
adding, subtracting, multiplying, and
Then the teacher calls on different
dividing rational numbers.
students and has them explain a
process from the Fractions without
Distractions notes.
The teacher then assigns several
different problems from the Basic Skills
Review Part I for students to explain
their processes using the notes from
Fractions without Distractions.
Phase: Evaluate
List of Materials:
Paper
Activity: Performance Assessment:
For each expression below, determine the solution and complete each
generalization.
1)
When adding two rational numbers, if a pair of addends has the same sign, then
the sum will have the __________ of both addends.
2)
When adding two rational numbers, if a pair of addends has the opposite signs,
then the sum will have the sign of the addend with the _________ ____________
_____________.
3)
When subtracting two non-zero positive rational numbers (with the subtrahend
larger than the minuend), the difference will always be ____________ than the
minuend and be ____________.
4)
When multiplying two non-zero positive rational numbers, where at least one of the
factors is greater than one, the product always will be __________ than the
smallest factor.
5)
When dividing two non-zero positive rational numbers, where the dividend is
greater than the divisor, the quotient will always be ___________ than one.
6)
When multiplying two or more rational numbers with a(n) __________ number of
negative signs, then the product is ___________________.
What’s the teacher doing?
What are the students doing?
The teacher is monitoring students as
they work.
The students are working individually to
complete the assignment.
Phase: Explore/Explain
List of Materials:
Buyer’s Bonanza
Activity: Buyer’s Bonanza
Distribute Buyer’s Bonanza to each student. Ask them to complete the tables.
What the teacher is doing
What the students are doing
The teacher is monitoring students as
they work, assisting as needed.
Students are working in pairs to
complete the handout.
How did you find the amount of
discount? How is that like finding the
sales tax?
What did you do to find the amount of
sales tax? How did you find the total
cost of the curtains? How is that
different from finding the amount of the
sale price?


What effect does the purchase price
of a given item have on sales tax?
What process is used to calculate
sales tax on a given purchase?
Phase: Explore/Explain
Materials:
You Can’t’ Hide from Taxes (one per student)
Activity: You Can’t Hide from Taxes
From the Texas Council on Economic Education lessons for Personal and
Financial Literacy for Grades 7 & 8, (free to all Texas educators on the internet
http://economicstexas.org/?page_id=5497 ), please distribute Lesson 1 “You
Can’t Hide from Taxes” and follow the lesson and teacher notes.
Phase: Elaborate
Materials:
Know Your Worth (one per student)
Activity:
From the Texas Council on Economic Education lessons for Personal and
Financial Literacy for Grades 7 & 8, (free to all Texas educators on the internet
http://economicstexas.org/?page_id=5497 ), please distribute Lesson 4 “Know
Your Worth” and follow the lesson notes.
Phase: Evaluate
List of Materials:
Paper
Activity: Performance Assessment:
Analyze the problem situation(s) described below. Organize and record your work
for each of the following tasks. Using precise mathematical language, justify and
explain each solution process.
Drew has decided it is time to purchase a new car. He has shopped around and
decided which car he would like to purchase.
1) The price of the car Drew wants to purchase is $17,998. This price includes all
fees associated with the purchase of a vehicle, such as title and registration fees;
however, the price does not include the required sales tax.
a) In Texas, the sales tax rate for automobiles is 6 %. Calculate the sales tax and
final price of the new car.
2) Drew will have to finance his car purchase through his bank with a car loan. The
financial institution requires him to provide a net worth statement to determine if he
will qualify for the loan. Drew’s current financial information is below:
a) Create and organize a financial assets and liabilities record and include a net
worth statement.
3) Drew’s car loan was approved for the purchase of a vehicle with monthly
payments that cannot exceed 12% of his current net worth.
a) Determine if the loan Drew was approved for will allow him to purchase his new
car, if his monthly car payment is projected to be $352.18.
4) Drew is going to file his 2014 federal income tax return and would like to use his
refund, if any, towards a down payment for the new car.
a) Using the taxable income brackets and rates from the Internal Revenue Service
below, calculate the income tax Drew is required to pay if:


his taxable income for 2014 was $48,750
he plans on filing as Head of Household
5) If the amount of paid income tax exceeds the required income tax, Drew will
receive a refund of the difference when he files his 2014 federal income tax return.
If the amount paid in income tax is less than the required income tax, Drew will
owe the difference when he files his federal income tax return. Drew paid a total of
$9,000 in income tax in 2014 through monthly deductions in his paycheck.
a) Determine if Drew will receive an income tax refund that he can use as a down
payment towards the purchase of his car and how this down payment will affect
the amount Drew will need to finance.
What is the teacher doing?
What are the students doing?
The teacher is monitoring students as
they work, allowing for individual
prompting, support and encouragement
through the task.
The student are working individually to
complete the task.
Engage Day 1
24
2
0
0.75
120%
−5
32
1
4
2
1
5
13
100
−0.25
3
2
50%
Explore 1: Day 1 Grouping Mat
Rational Numbers
Integers
Whole Numbers
Natural or Counting
Numbers
Notes on Rational Numbers
Sets of Numbers
Examples
Natural or Counting
Numbers
1, 2, 3, 4, 5, ….
Whole Numbers
0, 1, 2, 3, 4, 5, …
Integers
…-5, -4, -3, -2, -1, 0, 1,2 ,3 ,4 ,5 …
Rational Numbers
All natural or counting numbers
plus the whole numbers
plus the integers
plus the fractions where the numerator and
denominator are integers (positive and
negative)
plus the terminating and repeating decimals
plus the mixed numbers and decimals
Statement
All natural numbers are whole numbers.
All whole numbers are integers.
All integers are rational numbers.
All whole numbers are rational numbers.
All rational numbers are integers.
All integers are whole numbers.
All natural numbers are rational numbers.
The set of integers is a subset of the set of rational
numbers.
The set of natural numbers is a subset of the set of
whole numbers.
True or False?
Notes on Rational Numbers
Key
Statement
True or False?
All natural numbers are whole numbers.
True
All whole numbers are integers.
True
All integers are rational numbers.
True
All whole numbers are rational numbers.
True
All rational numbers are integers.
False
All natural numbers are rational numbers.
True
All integers are whole numbers.
False
The set of integers is a subset of the set of rational
numbers.
The set of natural numbers is a subset of the set of
whole numbers.
True
True
Another Look at Adding and Subtracting Fractions KEY
Problem
1.
2.
3.
1 3

8 8
1 3 4

8
8
44 1

84 2




3 1

4 3
3  3 1 4
9
4



4  3 3  4 12 12
94 5

12
12

3 2

5 3
3  3 2  5 9 10



5  3 3  5 15 15
9  10 19

15
15
19
4
1
15
15

4. 5 
5
3
5
3
5







Description of Process
Sample descriptions are given
Already has a common denominator
Add the numerators
Keep the common denominator
Simplify the resulting fraction
Rename each fraction with a common
denominator of 12
Subtract the numerators
Keep the common denominator
Rename each fraction with a common
denominator of 15
Add the numerators
Keep the common denominator
Write the improper fraction as a
mixed number
Does not need a common
denominator, since there is only one
fraction.
Combine the fraction with the whole
number to make a mixed number.
Another Look at Adding and Subtracting Fractions KEY
Problem
7 1
5. 2 
8 2
7 1 4
7 4
2 
2 
8 24
8 8
74
3
2
2
8
8
Description of Process
Sample descriptions are given
 Rename each fraction with a common
denominator of 8
 Combine the fractions by subtracting
the numerators and keeping the
common denominator
 Keep the whole number and combine
with the resulting fraction to form a
mixed number
3
8
3
8 3
2 1 
8
8 8
83
5
1
1
8
8

3 3
7. 1 
8 4
3 32
3 6
1 
1 
8 42
8 8
36
9
1
1
8
8
9
1
 1
1  1 1   2
8
8
 8

1 3
8. 2 
5 10
1 2 3
2
3
2

2 
5  2 10
10 10
12 3
 10 2  3
1 


1 

10 10
 10 10  10
12  3
9
1
1
10
10

6. 2 










Does not need a common
denominator
Rename the two as one and eighteighths so that three-eighths can be
subtracted.
Subtract the numerators
Keep the common denominator
Combine the fraction with the whole
number to make a mixed number.
Rename each fraction with a common
denominator of 8
Add the numerators
Keep the common denominator
Change the improper fraction to a
mixed number and combine it with
the whole number
Rename each fraction with a common
denominator of 10
Rename two and two-tenths as one
and twelve-tenths so that three-tenths
can be subtracted
Subtract the numerators
Keep the common denominator
Another Look at Adding and Subtracting Fractions KEY
9. Alice has two and one-fourth pies to eat. In one minute she ate six-eighths of a pie. How
much pie does Alice have left to eat?
1 6
2 6
10 6
4
1
2   2   1   1  1 pies
4 8
8 8
8 8
8
2
10. Charles ate one and one-half pies in 3 minutes and two-thirds of a pie in 2 minutes. How
much pie has Charles eaten in 5 minutes?
1 2
3 4
7
1
1
1   1   1  1  1  2 pies
2 3
6 6
6
6
6
11. Rachel has walked three-fifths of a mile. She has ten minutes to reach the two mile
marker. How far must Rachel walk to reach the two mile marker?
3
5 3
2
2   1   1 miles
5
5 5
5
12. What is the perimeter of the rectangle shown?
1
1
1
1
5
5
4
4
4 4 3 3  4
4
3
3
4
4
5
5
20
20
20
20
18
9
14
 14
feet
20
10
4 feet
3 feet
3
hours a day outside her home traveling to and from work and working
4
1
2
at her job. Kelly spends
hour one way traveling to work. Kelly is allowed
hour total
2
3
1
for her breaks. She is allowed 1 hour for lunch. The rest of the time Kelly spends
4
working in her office. How much time does Kelly spend working in her office?
3 1 1 2
1
9
11
21
11
10
5
9     1   9  2
8 2
6
 6 hours
4 2 2 3
4
12
12
12
12
12
6
13. Kelly spends 9
5
cm, how long is the base?
6
5  3
3
5
1
5
3
2
1
17   6  6   17  13  17  13  4  4 cm
6  4
4
6
2
6
6
6
3
14. If the perimeter of the isosceles triangle below is 17
cm
Another Look at Adding and Subtracting Fractions
Problem
1.
1 3

8 8
2.
3 1

4 3
3.
3 2

5 3
4. 5 
3
5
Description of Process
Another Look at Adding and Subtracting Fractions
Problem
7 1
5. 2 
8 2
6. 2 
3
8
3 3
7. 1 
8 4
1 3
8. 2 
5 10
Description of Process
Another Look at Adding and Subtracting Fractions
9. Alice has two and one-fourth pies to eat. In one minute she ate six-eighths of a pie. How
much pie does Alice have left to eat?
10. Charles ate one and one-half pies in 3 minutes and two-thirds of a pie in 2 minutes. How
much pie has Charles eaten in 5 minutes?
11. Rachel has walked three-fifths of a mile. She has ten minutes to reach the two mile
marker. How far must Rachel walk to reach the two mile marker?
12. What is the perimeter of the rectangle shown?
4 feet
3 feet
3
hours a day outside her home traveling to and from work and working
4
1
2
at her job. Kelly spends
hour one way traveling to work. Kelly is allowed
hour total
2
3
1
for her breaks. She is allowed 1 hour for lunch. The rest of the time Kelly spends
4
working in her office. How much time does Kelly spend working in her office?
13. Kelly spends 9
14. If the perimeter of the isosceles triangle below is 17
5
cm, how long is the base?
6
cm
Mac’s Brownies
Mac has created a new brownie, “Choco-Loco-Mint." One pan of brownies
yields 8 servings. (Note: 1 brownie represents 1 serving.) If Mac baked 3
1
2
pans of “Choco-Loco-Mint” brownies, how many brownies has he made?
Select one of the ranges of numbers below that could represent the solution:
(A)
(B)
(C)
20 to 30 brownies
24 to 30 brownies
24 to 32 brownies
Directions
Let the whole fraction square represent one pan.
Use the fraction squares to model the problem.
Draw a diagram of the fraction square model. Label all parts of the diagram
to communicate the solution.
Write an addition number sentence and a multiplication number sentence to
show the relationship between the numbers in the problem.
Mac’s Brownies KEY
Mac has created a new brownie, “Choco-Loco-Mint." One pan of brownies yields 8
1
servings. (Note: 1 brownie represents 1 serving.) If Mac baked 3 pans of “Choco2
Loco-Mint” brownies, how many brownies has he made?
Select one of the ranges of numbers below that could represent the solution.
(A)
(B)
(C)
20 to 30 brownies
24 to 30 brownies
24 to 32 brownies
Have students explain their selection. Keep a tally of all the student responses. This tally will be used to
compare to the reasonableness of the solution.
Directions
Let the whole fraction square represent one pan.
Use the fraction squares to model the problem.
Draw a diagram of the fraction square model. Label all parts of the diagram to communicate the
solution.
Write an addition number sentence and a multiplication number sentence to show the relationship
between the numbers in the problem.
1
We need to determine how many brownies are made with 3 pans (number of groups)
2
Each pan yields 8 brownies. (The number in each group)
I will lay 8 equal size pieces on each pan to represent the number of brownies each pan will yield.
1 pan yields 8 brownies
3
1
2
+
1 pan yields 8 brownies
+
pans of 8 brownies per pan = 3 pans of 8 brownies per pan +
1 pan yields 8 brownies
1
2
+
half pan yields 4 brownies
pan of 8 brownies =
24 brownies + 4 brownies = 28 brownies
Addition Number Sentence:
8 brownies + 8 brownies + 8 brownies + 4 brownies = 28 brownies
Multiplication Number Sentence:
3
1
2
pans x 8 brownies per/pan = 28 brownies
1
2
The multiplier (the first factor 3 ) indicates the number of times the multiplicand (the second factor 8) is used as an addend
when using repeated addition.
Dividing Trim
 Use the number line to model each given situation.
 Translate each situation to a division number sentence.
1. What is the maximum number of
1
yard lengths of ribbon trim that can be cut from a
4
3
yard length of ribbon trim?
4
3
yd.
4
0 yd.
1 yd.
Number Sentence:
2. What is the maximum number of
an
1
yard lengths of ribbon trim that can be cut from
12
8
yard length of ribbon trim?
12
0 yd.
8
yd.
12
1 yd.
Number Sentence:
3. What is the maximum number of
2
yard lengths of ribbon trim that can be cut from a
6
4
yard length of ribbon trim?
6
0 yd.
Number Sentence:
4
yd.
6
1 yd.
Dividing Trim
4. What is the maximum number of
a
3
yard lengths of ribbon trim that can be cut from
12
9
yard length of ribbon trim?
12
0 yd.
9
yd.
12
1 yd.
Number Sentence:
5. What is the maximum number of
2
yard lengths of ribbon trim that can be cut from a
6
5
yard length of ribbon trim?
6
5
yd.
6
0 yd.
1 yd.
Number Sentence:
6. What is the maximum number of
an
5
yard lengths of ribbon trim that can be cut from
12
8
yard length of ribbon trim?
12
0 yd.
Number Sentence:
8
yd.
12
1 yd.
Dividing Trim KEY
 Use the number line to model each given situation.
 Translate each situation to a division number sentence.
1. What is the maximum number of
1
yard lengths of ribbon trim that can be cut from a
4
3
yard length of ribbon trim?
4
1
yd
4
1
yd
4
1
yd
4
1 length
1 length
1 length
3
yd.
4
0 yd.
Number Sentence:
3
1
yd. 
yd. = 3 lengths
4
4
2. What is the maximum number of
an
1 yd.
1
yard lengths of ribbon trim that can be cut from
12
8
yard length of ribbon trim?
12
1
yd
12
1
length
1
yd
12
1
length
1
yd
12
1
length
1
yd
12
1
length
1
yd
12
1
length
1
yd
12
1
length
1
yd
12
1
length
1
yd
12
1
length
8
yd.
12
0 yd.
Number Sentence:
1 yd.
8
1
yd. 
yd. = 8 lengths
12
12
3. What is the maximum number of
2
yard lengths of ribbon trim that can be cut from a
6
4
yard length of ribbon trim?
6
1
yd.
6
1
yd.
6
1 length
1
yd.
6
1 length
4
yd.
6
0 yd.
Number Sentence:
1
yd.
6
4
2
yd. 
yd. = 2 lengths
6
6
1 yd.
Dividing Trim KEY
4. What is the maximum number of
a
3
yard lengths of ribbon trim that can be cut from
12
9
yard length of ribbon trim?
12
1
yd
12
1
yd
12
1
yd
12
1
yd
12
1 length
1
yd
12
1
yd
12
1
yd
12
1 length
1
yd
12
1
yd
12
1 length
9
yd.
12
0 yd.
Number Sentence:
1 yd.
9
3
yd. 
yd. = 3 lengths
12
12
5. What is the maximum number of
2
yard lengths of ribbon trim that can be cut from a
6
5
yard length of ribbon trim?
6
1
yd.
6
1
yd.
6
1
yd.
6
1 length
1
yd.
6
1
yd.
6
1
2
1 length
5
yd.
6
0 yd.
Number Sentence:
1 yd.
5
2
1
yd. 
yd. = 2 lengths
6
6
2
6. What is the maximum number of
an
length
5
yard lengths of ribbon trim that can be cut from
12
8
yard length of ribbon trim?
12
1
yd
12
1
yd
12
1
yd
12
1 length
1
yd
12
1
yd
12
1
yd
12
1
yd
12
3
5
length
8
yd.
12
0 yd.
Number Sentence:
1
yd
12
8
5
3
yd. 
yd. = 1 lengths
12
12
5
1 yd.
Extending Integers
1. Use the given clues to identify the three integers. Write an equation to show the sum of the three
integers and use a number line to model the equation.
Clue 1: The first integer is 2 units to the left of 0.
Clue 2: The second integer is 2 units less than the first integer.
Clue 3: The third integer is 6 units more than the first integer.
2. Use the given clues to identify the three integers. Write an equation to show the difference of the
three integers and use a number line to model the equation.
Clue 1: The first integer is 2 more than negative 7.
Clue 2: The second integer is 4 less than 3.
Clue 3: The third integer is 6 units more than the first integer.
3. Use the given clues to identify the three integers. Write an equation to show the product of the three
integers.
Clue 1: The first integer is 4 less than negative 6.
Clue 2: The second integer is 4 less than 23.
Clue 3: The third integer is 8 units more than the first integer.
4. Use the given clues to identify the two integers. Write an equation to show the quotient of the two
integers and use positive and negative tiles to model the equation.
Clue 1: The first integer is 14 more than negative 6.
Clue 2: The second integer is 4 less than 2.
Extending Integers
Write an equation and use a vertical number line to match the given situation for problems 5 through 8.
5. An elevator was on
the third floor and
went down four
floors. Where is the
elevator?
3
2
7. The elevator was on
the first floor and
was needed on the
second floor down
in the basement.
How many floors
did it need to move
and in what
direction?
6. The elevator was
two floors down in
the basement and
went up five floors.
Where is the
elevator?
3
2
1
1
0
0
–1
–1
–2
–2
–3
–3
3
2
1
8. The elevator was 1
floor down in the
basement and was
needed on the
second floor. How
many floors did it
need to move and in
what direction?
3
2
1
0
0
–1
–1
–2
–2
–3
–3
Extending Integers
9. Gary was reviewing his monthly bank statement. He had three withdrawals of $75, $150, and $425.
He had two deposits of $500 and $206. If his balance at the beginning of the month was $29, what
is his current balance after the withdrawals and deposits? Write a number sentence to match the
situation and solve the problem.
Extending Integers KEY
1. Use the given clues to identify the three integers. Write an equation to show the sum of the three
integers and use a number line to model the equation.
Clue 1: The first integer is 2 units to the left of 0.
Clue 2: The second integer is 2 units less than the first integer.
Clue 3: The third integer is 6 units more than the first integer.
Answer: first integer is (−2); second integer is (−4) = (−2) + (−2); third integer is (−2) + 6 = 4
Equation: (−2) + (−4) + 4 = (−2)
–7 –6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
7
2. Use the given clues to identify the three integers. Write an equation to show the difference of the
three integers and use a number line to model the equation.
Clue 1: The first integer is 2 more than negative 7.
Clue 2: The second integer is 4 less than 3.
Clue 3: The third integer is 6 units more than the first integer.
Answer: first integer is (−5) = 2 + (−7); second integer is (−1) = 3 + (−4); third integer is (−5) + 6 = 1
Equation: (−5) ─ (−1) ─ 1 = (−5); Since 1 and (−1) create a zero pair, they do not count as an overlap on the
number line.
opposite of 1
opposite of (−1)
–7 –6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
7
3. Use the given clues to identify the three integers. Write an equation to show the product of the three
integers.
Clue 1: The first integer is 4 less than negative 6.
Clue 2: The second integer is 4 less than 23.
Clue 3: The third integer is 8 units more than the first integer.
Answer: first: (−10) = (−6) + (−4); second: 19 = 23 + (−4); third: (−2) = (−10) + 8
Equation: (−10)(19)(−2) = 380
4. Use the given clues to identify the two integers. Write an equation to show the quotient of the two
integers and use positive and negative tiles to model the equation.
Clue 1: The first integer is 14 more than negative 6.
Clue 2: The second integer is 4 less than 2.
Answer: first: 8 = (− 6) + 14 second: (−2) = 2 + (−4)
Equation: 8 ÷ (−2) = (−4)  Divide 8 positive tiles equally into 2 rows and then take the opposite of the
tiles in each row.
(+)
(+)
(+)
(+)
(+)
(+)
(+)
(+)
(+)
(+)

(–)
(–)
(–)
(–)
(–)
(–)
(–)
(–)

(+)
(+)
(+)
(+)
(+)
(+)
Extending Integers KEY
Write an equation and use a vertical number line to match the given situation for problems 5 through 8.
5. An elevator was on
the third floor and
went down four
floors. Where is the
elevator?
Answer: 3 + (−4) =
(−1)
7. The elevator was on
the first floor and
was needed on the
second floor down
in the basement.
How many floors
did it need to move
and in what
direction?
You want the distance
between (−2) and 1, so
you subtract.
Answer: (−2) ─ 1 =
(−3)
(−2) + (−1) = (−3)
3 floors down.
3
2
6. The elevator was
two floors down in
the basement and
went up five floors.
Where is the
elevator?
Answer: (−2) + 5 = 3
3
2
1
1
0
0
–1
–1
–2
–2
–3
–3
3
2
1
0
–1
–2
–3
8. The elevator was 1
floor down in the
basement and was
needed on the
second floor. How
many floors did it
need to move and in
what direction?
You want the distance
between 2 and (−1), so
you subtract.
Answer: 2 ─ (−1) = 3
2+1=3
3 floors up.
3
2
1
0
–1
–2
–3
Extending Integers KEY
9. Gary was reviewing his monthly bank statement. He had three withdrawals of $75, $150, and $425.
He had two deposits of $500 and $206. If his balance at the beginning of the month was $29, what
is his current balance after the withdrawals and deposits? Write an equation to match the situation
and solve the problem.
Answer: $29 ─ 75 ─ 150 ─ 425 + 500 + 206 = $85
Basic Operations Notes
Fractions
Multiplication:
To multiply fractions, you multiply straight across.
1
3
3

=
Example:
8
8
64
In some cases you might need to simplify the answer, so make sure you double check your answer.
You can simplify the product after multiplying Example:
2
1
2 2
1

=
( is simplified to )
3
2
6 6
3
Or you can simplify the factors before multiplying Example:
4
1
2
1
4
2
2
1
2

becomes

because ( is simplified to ), so

=
6
5
3
5
6
3
3
5
15
If one of the factors is a whole number, you write the whole number as an improper fraction by
writing the whole number as the numerator and by placing a “1” in the denominator.
1
3
1
3
becomes

=
Example: 3 
5
1
5
5
If one of the factors is a mixed number, you write the mixed number as an improper fraction.
2
1
17
1 17
2
becomes

=
=1
Example: 3 
5
3
5
3
15
15
(To write a mixed number as an improper fraction, you multiply the whole number by the
denominator, and then add the numerator. In the problem above, multiply 5 times 3 then add 2,
keeping the original denominator.)
Division:
To divide fractions, you keep the first term the same, change the division to multiplication, then flip
the second term. Another way of saying the same thing is to multiply by the reciprocal of the
second term. If either term is a mixed number or whole number, you will use the procedures given
above in the section on multiplication.
Example:
5
1
5
2 10
5
1

=

=
. This answer simplifies to
=1 .
8
2
8
1
8
4
4
Example: 4
2
1
22
3
66
66
1

becomes

=
and
= 13 .
5
3
5
1
5
5
5
Basic Operations Notes
Addition and Subtraction (fractions):
To add or subtract fractions you must have a common denominator.
5
1
1
2
+
, the least common demoninator is 8.
can be written as .
8
2
2
8
Example:
5
2
7
So,
+
= .
8
8
8
For
13
5
13
39

, the least common demoninator is 9.
can be written as
.
3
9
3
9
Example:
39
5
34
34
7
So,

=
. This solution may be written as
or 3 .
9
9
9
9
9
For
Decimals
Addition and Subtraction:
The decimals in the problem must have the place values aligned to add or subtract. If the problem is
written across the page, it is recommended you rewrite the problem vertically to perform the
addition or subtraction.
Example: 2.345 + 17.4 would be written as: 2.345
+17.4
Example:
62.34
+51.4
0.0135
+ 3.09
21.897
4.5060
– 0.645
–1.3450
Multiplication:
Complete the multiplication, then place the decimal in the answer based on the number of digits to
the right of the decimal in each factor. The decimals in the problem do not have to line up.
Example:
1.234 (there are 3 digits to the right of the decimal)
X 5.67 (there are 2 digits to the right of the decimal)
-------6.99678 (there are 5 digits to the right of the decimal)
Example: 5.4(3.02) = 16.308 (There is one digit to the right of the decimal in the first factor
plus two digits to the right of the decimal in the second factor, which makes three digits to the
right of the decimal in the product.)
Decimal Division:
When dividing a decimal by a whole number, the decimal in the product must be lined up with the
1 .4
decimal in the quotient.
Example: 4.2 ÷ 3 = 1.4
 3 4 .2
When dividing a decimal by a decimal, you must move the decimals in the dividend and divisor the
number of places required to make the divisor a whole number.
Example: 24.9 ÷ 0.3 you will need to move the decimals in each
number exactly one digit. The problem becomes 249 ÷ 3.
0.3 24.9

3 249
0.03 24.9

3 2490
It might be necessary to add one or more zeros as you are moving decimals.
Example: 24.9 ÷ 0.03. Each decimal must be moved two digits. The
problem becomes 2490 ÷ 3.
Basic Operations Notes
Integers
Multiplication and Division:
If the signs are the same, the answer is positive.
If the signs are different, the answer is negative.
Example: (24)(3) = 72
(−108)(−4) = 432
(24)(−3) = (−72)
(−108)(4)= (−432)
If the problem has more than two integers, you begin with the first two integers and perform the
multiplication or division using the rule above. Then, using the sign of that product perform the
next operation.
Example: (−2)(3)(4)  (−2)(3) = (−6)  (−6)(4) = (−24)
Addition:
When adding integers that have the same sign, just add the integers and keep the sign.
Example: 4 + 9 = 13 or (−4) + (−9) = (−13)
If the signs are NOT the same, find the difference between the two numbers and use the sign of the
number that is the greatest distance from zero.
Example: (−5) + 4 = (−1) and 12 + (−3) = 9
Subtraction:
To subtract an integer you will need to add the opposite of the integer being subtracted.
Example: 7 – (−3) first change the sign from subtraction to addition, then change the (−3) to
(+3). The problem becomes 7+ (+3) = 10.
Example: (−12) – 8 would become (−12) + (−8).
It is important to remember that the first term does not change its sign!
Basic Operations Skill Drills Part 1 KEY
Integers
Simplify:
1) -102+ 465 = 363
2) 365 + (-241) = 124
3) (-18) + (-16) = (-34)
4) 3657 – (-1245) = 4902
5) -357 – (-291) = (− 66)
6) (-487) -1357 = (-1844)
7) (13)(15) = 195
8) (-144)(12) = (-1728)
9) (- 60)(- 50) = 3000
10) (21)(-17) = (-357)
11) 468 ÷ (-12) = (-39)
12) ( -678) ÷ (-226) = 3
13)
-575
1
= (- )
1150
2
14) (15)(-27) ÷(-5) = 81
Basic Operations Skill Drills Part 1 KEY
Fractions
Simplify:
1.
5
1
2
1

=
=
6
2
6
3
2.
2
4
8

=
5
5
25
3.
7
2
11
1

=
or 1
10
5
10
10
4.
1
1
 2=
2
4
4 3
12
6
5. ( )( ) =
or
5 14
70
35
6.
3
1
13

=
8
6
24
3
4
15
7. (- )  (- ) =
4
5
16
8.
3
5
3

=
4
4
5
8
2
14
5
9. (- ) 
= (- ) or -1
9
3
9
9
10.
4
2
8
4

=
or
6
9
18
9
2
3
6
2
11. (- ) 
= (- ) or (- )
3
5
15
5
12. 5  3
5
19
3
13. -4  (-1 ) = (- ) or (-2 )
8
8
8
1
2
14. 2 =
or 1
1
2
2
1
7
3
=
or 1
4
4
4
Basic Operations Skill Drills Part 1
Integers
Simplify:
15) -102+ 465 =
16) 365 + (-241) =
17) (-18) + (-16) =
18) 3657 – (-1245) =
19) -357 – (-291) =
20) (-487) -1357 =
21) (13)(15) =
22) (-144)(12) =
23) (-60)(-50) =
24) (21)(-17) =
25) 468 ÷ (-12) =
26) ( -678) ÷ (-226) =
-575
=
1150
28) (15)(-27) ÷(-5) =
27)
Basic Operations Skill Drills Part 1
Fractions
Simplify:
15.
5
1

=
6
2
16.
2
4

=
5
5
17.
7
2

=
10
5
18.
1
 2=
2
4 3
19. ( )( ) =
5 14
20.
3
1

=
8
6
3
4
21. (- )  (- ) =
4
5
22.
3
5

=
4
4
8
2
=
23. (- ) 
9
3
24.
4
2

=
6
9
2
3
=
25. (- ) 
3
5
26. 5  3
5
27. -4  (-1 ) =
8
1
28. 2 =
1
2
1
=
4
Basic Operations Skill Drills Part 2 KEY
Decimals
Simplify:
29) 8.3 + 0.24 + 6 = 14.54
30) 46 – 18.9 = 27.1
31) (0.21)(0.3) = 0.063
32) 5 – 2.4 – 1.38 + 0.62 = 1.84
33) 33.7 – 98.68 = (-64.98)
34) (1.3)(0.005) = 0.0065
35) (-2.5)(0.012) = (-0.03)
36) 7.026 ÷ (-0.03) = (-234.2)
37)
-8.34
= 4,170
-0.002
38) 13.05 ÷ 0.4 = 32.625
39) (-2.5) – 3.4 + 2 = (-3.9)
40) 5 – (-2.5) – 9 = (-1.5)
41) 4 + 3.08 – 0.99 = 6.09
42) 3.2 ÷ 0.002 = 1,600
Basic Operations Skill Drills Part 2 KEY
Mixed Review
Simplify:
1)
7
1
35
3
 1 =
or 1
8
4
32
32
3)
4
3
3
437
45
 3 =
or 7
8
7
56
56
2)
5
3
20
5

=
or
8
4
24
6
4)
5
3
1
25
1
 2 =
or 3
8
4
8
8
5) 8% sales tax for $24.58
(0.08)(24.58) = 1.9664
6) 152.5 ÷ 0.05 = 3050
7) 5 – 2.57 + 8.623 = 11.053
8) Half off $189.98
1
189.98 •
= 94.99
2
9) (-50) (-4) = 200
10)
11) (-18) + (-7) = (-25)
12) (-202) – 81 = (-283)
1
13) (4.03)( (- ) ) = (-2.015)
2
14) -52.31 – 48.224 = (-100.534)
-305
= 61
-5
Basic Operations Skill Drills Part 2
Decimals
Simplify:
43) 8.3 + 0.24 + 6 =
44) 46 – 18.9 =
45) (0.21)(0.3) =
46) 5 – 2.4 – 1.38 + 0.62 =
47) 33.7 – 98.68 =
48) (1.3)(0.005) =
49) (-2.5)(0.012) =
50) 7.026 ÷ (-0.03) =
51)
-8.34
=
-0.002
52) 13.05 ÷ 0.4 =
53) (-2.5) – 3.4 + 2 =
54) 5 – (-2.5) – 9 =
55) 4 + 3.08 – 0.99 =
56) 3.2 ÷ 0.002 =
Basic Operations Skill Drills Part 2
Mixed Review
Simplify:
15)
7
1
 1 =
8
4
17) 4
3
3
 3 =
8
7
16)
5
3

=
8
4
18) 5
3
1
 2 =
8
4
19) 8% sales tax for $24.58
(0.08)(24.58) =
20) 152.5 ÷ 0.05 =
21) 5 – 2.57 + 8.623 =
22) Half off $189.98
1
189.98 •
=
2
23) (-50)(-4) =
24)
25) (-18) + (-7) =
26) (-202) – 81 =
1
27) (4.03)( (- ) ) =
2
28) -52.31 – 48.224 =
-305
=
-5
Buyer’s Bonanza
Bonnie’s mother bought curtains on sale. The regular price of the curtains was $76. The curtains
were marked 25% off. Use the percent bar model below to show the amount of the discount and the
sale price of the curtains.
Dollars
0%
25%
50%
75%
100%
Percent
Regular Price of
Curtains
Amount of 25% Discount
Sale Price
How Sales Tax Works
Bonnie’s mother bought curtains. The sales tax on the price of the curtains was 5%.
Use the percent bar model below to show the amount of the sales and the total cost of the curtains.
Dollars
0%
25%
50%
75%
100%
Percent
Price of Curtains
Amount
of Sales
Tax
Total Cost of
Curtains
Buyer’s Bonanza
Key
Bonnie’s mother bought curtains on sale. The regular price of the curtains was $76. The curtains
were marked 25% off. Use the percent bar model below to show the amount of the discount and the
sale price of the curtains.
Dollars
$0
$19
$57
$38
Amount of discount
0%
25%
$76
Sale price
50%
75%
100%
Percent
Regular Price of
Curtains
Amount of 25% Discount
Sale Price
$76
$19
$57
How Sales Tax Works
Bonnie’s mother bought curtains. The sales tax on the price of the curtains was 5%.
Use the percent bar model below to show the amount of the sales and the total cost of the curtains.
Dollars
$0
$2.85
$57
Tax
0%
25%
5%
50%
75%
100%
$59.85
$0
$2.85
$57
Tax
0%
5%
25%
50%
75%
100%
105%
Percent
Price of Curtains
Amount
of Sales
Tax
Total Cost of
Curtains
$57
$2.85
$59.85