Asbury Park Middle School
Accentuate the Negative
Department: Math Unit designation: #4
Course: 7 Anticipated timeframe: approx. 4 weeks
Standards addressed: 7 RP 1,2,3 NS 1,2,3 EE 3,4
Analyze proportional relationships and use them to solve real-world
and mathematical problems.
1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities
measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit
rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
2. Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a
table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal
descriptions of proportional relationships.
c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number
n of items purchased at a constant price p, the relationship between the total cost and the number of items can
be expressed as t = pn.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with
to the points (0, 0) and (1, r) where r is the unit rate.
3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest,
tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
The Number System 7.NS
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and
divide rational numbers.
1. Apply and extend previous understandings of addition and subtraction to add and subtract rational
numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0
charge because its two
constituents are oppositely charged.
b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction
depending on whether q is
positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret
sums of rational
numbers by describing real-world contexts.
c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the
two rational numbers on the number line is the absolute value of their difference, and apply this principle in
d. Apply properties of operations as strategies to add and subtract rational numbers.
2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and
divide rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations
satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1)
= 1 and the rules
for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of
integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q).
Interpret quotients of rational numbers by describing real world contexts.
c. Apply properties of operations as strategies to multiply and divide rational numbers.
d. Convert a rational number to a decimal using long division; know that the decimal form of a rational
number terminates in 0s or
3. Solve real-world and mathematical problems involving the four operations with rational numbers.1
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in
any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations
to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness
of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour
gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50.
If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches
wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on
the exact computation.
4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple
equations and inequalities to solve problems by reasoning about the quantities.
a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are
specific rational numbers.
Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying
the sequence of the
operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What
is its width?
and describe the solutions.
Transfer Goals: To select and use strategies to:
Use positive and negative numbers to graph in four quadrants and to model and answer questions about
Compare and order rational numbers Understand relationships between positive and negative numbers
Develop algorithms for all operations using positive and negative numbers
Enduring Understandings: The learner will Essential Questions:
understand that: -What is the effect of negative and positive numbers
-Relationships exist between positive and negative on operations?
integers. -In what professions would the use of negative and
- Operations can be used to solve problems and positive integers be an integral part of the job
equations with both positive and negative numbers requirements?
- How would you feel if you were on the losing team
in Math Mania? How would you feel if you were the
person who answered the question incorrectly?
-How do you locate points and lines on a coordinate
grid using all four quadrants?
Learners will know: Learners will be able to:
To use appropriate notation to indicate positive Compare and order rational numbers
and negative numbers How to locates rational Understand the relationship between positive and
numbers on a number line negative numbers
Order of operations Develop algorithms for all four operations with
Commutative property negative numbers
Distributive property Write mathematical sentences to show
Additive inverse relationships
Use positive and negative numbers to graph in
four quadrants and to model and answer
questions about applied settings
Performance Tasks: Other Evidence:
Unit openers Informal observations
Mathematical Highlights Portfolio assessment
Unit projects Homework assignments
Unit “Investigations” Classroom discussions using Socratic
Calculator activities questioning techniques
“Launching” the unit activities
Journal writing on each unit
“Cooperative learning” assignments
Pre and post assessments per each unit
Open ended questions
Constructed response questions
1. What is the order of operations? Why is this important for you to understand?
Give an example where the use of parentheses changes the result of the computation
2. You are to balance your checkbook.
- You begin with $25.00
- You make a deposit of $76.00
- You write a check for 2 shirts that cost $12.50 each
- You deposit $10.00 You purchases sneakers for $119.00
What is your balance?
- You want to purchase a video game for $25.00 Do you have enough money? How much extra, or how
much more money do you need?
3. The following shows yards lost and gained on each play by our football team
-8, 20, 3, 7, 15, 4, -12, 32, 5, 1
Write an expression that shows how to compute their average gain or loss per play. Then compute the
Anticipated daily sequence of activities:
Identify prime and composite number
Simply: 3 x 10/2 =
Solve: ¾ - 2/3
Order of operations
Locate numbers on a number line (both positive and negative)
Identify: Commutative Property of addition and multiplication, Associative Property of Addition and
Multiplication and Distributive Property
As students progress through this unit, they should be asked the following questions to assess their
specific knowledge of the unit.
How do positive and negative numbers help in describing the situation?
What will addition subtraction, multiplication, or division of positive and negative numbers tell about the
What model(s) for positive and negative numbers would help in showing the relationships in the problem
How can you tell which number is bigger?
Where is that number located on a number line?
How could that number be represented on a chip board?
Does your answer make sense?
Can you put this information in terms of “absolute value”?
Should the answer be positive, negative, or zero?
Extending the Number System
Focus: Compare and apply integers using symbols (< = >)
Activity: Fold a strip of paper in half. Label zero. Use rulers to place integers. Discuss “unit size” and
scale. Use student number lines to add integers.
Playing math Fever (1.1) Revision: Use playing cards. Pair/Share. Worksheet for keeping track. Have
students play “war”. One player is assigned to be positive and the second player is the negative. Add
the integers. Write the problem on the worksheet along with solution. Must complete at least 10
Activity 1.2 Have students turn their number lines vertically to help solve the problems pgs. 10-13.
Applications pgs. 16-17.
Extensions pg. 20 #48 Lab sheet available.
Writing: Reflections pg. 21
Adding and Subtraction Integers
#1 Focus: Understanding Absolute Value…Sometimes you only want the distance and not the direction.
#2 Focus: Develop an understanding between addition and subtraction of integers.
Introduction: pg. 26 Using a number line.
Activity: Using student number lines, complete 2.2 pg. 27
Activity: Writing Equivalent problems using addition and subtraction.
Addition and Subtraction Relationships pg. 29
Introduce algorithms. (rules for adding & subtracting integers) Students should test each
algorithm with a number line.)
Students need to understand the difference between a “negative sign” and a “subtraction sign”.
Applications: pg. 32--#1-#3 & pg. 33 #6-#8.
Connections: pg. 38 --#31-#34.
Writing: Reflections pg. 41.
Multiplying and Dividing Integers
Focus: Learn to multiply and divide integers using algorithms.
Activity: Integer Product Game. Pg. 49 Use this board to practice both multiplication and division of
integers. Lab sheet 3.4
Activity: Applications: pg. 51-52
Activity: Connections: pg. 53 #24
Writing: Reflections pg. 59
Properties of Operations
#1-Focus: Distributive Property
#2- Focus: Order of Operations
(two strategies to solve area problems)
Note: Distributive Property is going to require time and practice. It is a foundation for Grade 8 math
and again in 9th grade algebra.
Activity: 4.2 pg. 64-65
Activity: 4.3 pg. 67
Applications: pg. 69-70
Connections pg. 70-71
Writing: Reflections pg. 75
Vocabulary: counting numbers, whole numbers, integers, positive numbers, negative numbers, rational
numbers, absolute value, order of operations, distributive property, algorithm
Connected Mathematics units: Prime Time, Bits and Pieces I, II, III and Covering and Surrounding
Test preparation materials, such as Buckle Down
Notes and definitions:
Standards are obtained from the Common Core standards adopted by NJDOE in June 2010.
Transfer goals are long term reasons why students should learn the information for use in their lives as
Enduring Understandings are generalized, big picture ideas beyond specific content that describe the
realizations learners are to take from this unit into their lives functioning as adults. Begins with “The
learner will understand that . . .”
Essential Questions are to be motivational for students, derived from the Enduring Understandings, and
provide a basis for closure at the end of the unit.
Learners will know describes the concepts to be learned in the chapter
Learners will be able to describes the skills the learners will perform to be successful
Performance Tasks are the activities learners will undertake to demonstrate proficiency, e.g. tests,
quizzes, projects, Do Nows, homework . . .
Other Evidence may describe self assessments, homework review, employment of rubrics for scoring,
other similar practices.
Authentic Assessment is a description of the authentic assessment activity students will undertake in
the unit. “Authentic” describes an application of the skills and concepts learned to that point in the unit
that gives the learner a real life task they would face in a work or life related role.
Anticipated daily sequence of activities describes the progression of daily topics that will be further
detailed in the daily lesson plans. Included in each item/day should be, at least, new material to be
learned and previously learned material to be reviewed.
Anticipated resources list those that are expected to be useful or required for the learning activities
during the unit.