Download Unit 2 - Rational and Irrational Numbers

7/8 Compacted Mathematics Curriculum Guide
2016 – 2017
Unit 2: Rational Numbers – No Calculators
Time Frame: Quarter 1 – about 15 days
Connections to Previous Learning:
In Grade 6, students develop an understanding of rational numbers by using vertical and horizontal number lines and by applying their previous knowledge of
whole numbers and integers to rational number systems.
Students have a great deal of background in operations from K-6 that will lead to achievement of the expectations in this unit. In the primary grades, students
build understanding of whole number addition and subtraction. In the intermediate grades, students build understanding of multiplication and division with
whole numbers and addition, subtraction, multiplication and division with fractions and decimals. 6th grade students learn to divide fractions by fractions with
understanding and work to build an understanding of the value of integers. All of these concepts are crucial for students to be successful in this unit on
operations with rational numbers. In the past, students have used number line models with whole numbers, fractions and decimals. 6th graders have
represented integers on vertical and horizontal number line diagrams, but this is their first experience in adding and subtracting positive and negative rational
numbers.
Students in Grade 7 learn to differentiate between terminating and repeating decimals. In Grade 8, students realize that terminating decimals are repeating
decimals that repeat the digit zero. Students in Grades 6 and 7 have learned to use expressions, equations and inequalities to represent problem solving
situations.
Students in Grade 8 will expand upon those skills to include work with very large and very small numbers involving the use of integer exponents.
Focus within the Grade Level:
Students will be able to add, subtract, multiply, and divide rational numbers fluently as well as solve real-world and mathematical problems. In this unit, students
will represent their calculations using number line models, equations or expressions, and real world applications. Students will understand the value of square
roots and cube roots and use this understanding to solve equations involving perfect squares and cubes. Further work with exponents, including scientific notation,
naturally flow from the understanding of squares and cubes.
Connections to Subsequent Learning:
Students will use their understanding of rational numbers and apply them to linear functions. Solving equations of the form 𝑥 2 = p reminds students about
inverse operations which they will need to explore the topics of solving linear and proportional equations later in the year. Exponents and roots also connect
closely to work with the Pythagorean Theorem and volume of rounded objects later in Grade 8.
Mathematical Practices
1. Make Sense of Problems and Persevere in Solving Them.
2. Reason Abstractly and Quantitatively.
3. Construct Viable Arguments and Critique the Reasoning of Others.
4. Model with Mathematics.
Unit 2
5. Use Appropriate Tools Strategically.
6. Attend to Precision.
7. Look for and Make Use of Structure.
8. Look for and Express Regularity in Repeated Reasoning
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2016 – 2017
Stage 1 Desired Results
Transfer Goals
Students will be able to independently use their learning to…

Apply problem solving strategies to scenarios involving rational numbers in the marketplace.
Meaning Goals
UNDERSTANDINGS
Students will understand that…
 Rational numbers use the same properties as whole numbers.
 Rational numbers can be used to represent and solve real-life
situation problems.
 Rational numbers can be represented with visuals (including
distance models), language, and real-life contexts.
 A number line model can be used to represent the unique
placement of any number in relation to other numbers.
 There are precise terms and sequence to describe operations
with rational numbers.
ESSENTIAL QUESTIONS
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How are rational numbers used and applied in real-life and mathematical
situations?
What is the relationship between properties of operations and types of numbers?
Acquisition Goals
Students will know…
 How to define absolute value.
 Adding in zero pairs of integer chips does not change the total.
 Values can be expressed as positive and negative decimals,
fractions, and mixed numbers.
 A number next to a value in parenthesis symbolizes
multiplication.
 The decimal form of a rational number terminates in zeros or
eventually repeats.
 Properties:
o additive inverse
o associative
Unit 2
Students will be skilled at…
 Add and subtract rational numbers. (7.NS.A.1)
 Represent addition and subtraction on a horizontal or vertical number line
diagram. (7.NS.A.1)
 Use words, visuals and symbols to describe situations in which opposite quantities
combine to make 0. (7.NS.A.1)
 Represent addition of quantities with symbols, visuals and words by showing
positive or negative direction from one quantity to the other. (7.NS.A.1)
 Show that a number and its opposite have a sum of 0 using visuals, symbols,
words and real-world contexts. (7.NS.A.1)
 Use the term “additive inverse” to describe 2 numbers whose sum is zero.
(7.NS.A.1)
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7/8 Compacted Mathematics Curriculum Guide
o
o
commutative
distributive
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Calculators
7.NS.A.1
7.NS.A.1a
7.NS.A.1b
7.NS.A.1c
7.NS.A.1d
7.NS.A.2
7.NS.A.2a
7.NS.A.2b
7.NS.A.2c
Unit 2
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2016 – 2017
Use commutative, distributive, associative, identity, and inverse properties to add
and subtract rational numbers. (7.NS.A.1)
Use the term “absolute value” to describe the distance from zero on number line
diagram and with symbols. (7.NS.A.1)
Multiply and divide rational numbers. (7.NS.A.2)
Use the distributive property to multiply positive and negative rational numbers
using symbols, visuals, words and real-life contexts. (7.NS.A.2)
Interpret products of rational numbers by describing real-world contexts.
(7.NS.A.2)
Identify situations when integers can and cannot be divided. (7.NS.A.2)
Use words and real-world contexts to explain why the quotient of two integers is a
rational number. (7.NS.A.2)
Identify and apply properties used when multiplying and dividing rational
numbers.
(7.NS.A.2)
Convert a rational number to a decimal using long division. (7.NS.A.2)
Identify terminating or repeating decimal representations of rational numbers.
(7.NS.A.2)
Solve real world and mathematical problems involving the four operations with
rational numbers. (7.NS.A.3)

Materials Needed for Unit
Holt Course 2
Holt Course 3
Additional Materials
CMP 2 Accentuate the Negative
Prerequisite Skills Required for this Unit
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no
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7.NS.A.2d
7.NS.A.3
2016 – 2017
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Stage 1 Established Goals: Common Core State Standards for Mathematics
Number System
7.NS.A Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
7.NS.A.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.
7.NS.A.1a 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.
7.NS.A.1b 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.
7.NS.A.1c Understand subtraction of rational numbers as adding additive inverse, p- q = p + (-q). Show that the distance between two rational
numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
7.NS.A.1d Apply properties of operations as strategies to add and subtract rational numbers.
7. NS.A.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
7. NS.A.2a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to 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.
7. NS.A.2b 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/1) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts.
7. NS.A.2c Apply properties of operations as strategies to multiply and divide rational numbers.
7. NS.A.2d Convert a rational number to a decimal using long division; know that the decimal form of a rational number
7. NS.A.3 Solve real-world and mathematical problems involving the four operations with rational number.
 Major Clusters (approximately 85% of instruction)
Suggested Assessments
See Sample Assessments for Unit 2.
Unit 2
 Supporting Clusters
 Additional Clusters
Fluency Activities
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2016 – 2017
Select Activities from math-grades-6-8-fluency-support.pdf as is appropriate.
Teachers may start the year doing fluency activities from 6th grade
standards.
Vocabulary
Absolute Value (6) - The distance of a number from zero on a number line. Shown by l l.
Additive Inverse (6) -Two numbers whose sum is 0 are additive inverses of one another. Example: 3/4 and – 3/4 are additive inverses of one another
because 3/4 + (- 3/4) = (- 3/4) + 3/4 = 0.
Associative Property (3) - The property that states you can add or multiply regardless of how the numbers are grouped.
Commutative Property (3) - The property that states that two or more numbers can be added or multiplied in any order without changing the outcome.
Difference (K) -The result when one number is subtracted from another.
Distributive Property (3)-The Distributive Property of Multiplication is the property that states that multiplying a sum by a number is the same as
multiplying each addend by the number and then adding the products.
Expressions (K)- A mathematical phrase that contains operations, numbers, and/or variables.
Income/Profit (6)- money received, especially on a regular basis, for work or through investments/ the difference between the amount earned and the
amount spent in buying, operating, or producing something
Integer (6)-A number expressible in the form a or -a for some whole number a.
Negative Numbers (6) - Numbers less than zero
Opposite (5) Two numbers that are an equal distance from zero on a number line.
Positive Numbers(6) -Numbers greater than zero
Rational Numbers (6) -A number expressible in the form a/b or – a/b for some fraction a/b. The rational numbers include the integers.
Repeating Decimals (6) - A decimal in which one or more digits repeats infinitely.
Sum (K)-The result when two or more numbers are added.
Terminating Decimal (6)-A decimal is called terminating if its repeating digit is 0.
7.NS.A.1
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2016 – 2017
Vocab – Rational Numbers, Associative Property, Commutative Property, Distributive Property, Negative Numbers, Opposites, Positive Numbers,
Income/Profit, Absolute Value, Additive Inverse, Number Line, operations,
Continued on next page……
Visual representations may be helpful as students begin this work; they become less necessary as students become more fluent with the operations.
Examples:
 Use a number line to illustrate:
 p-q
 p + (- q)
 Is this equation true p – q = p + (-q)
 -3 and 3 are shown to be opposites on the number line because they are equal distance from zero and therefore have the same absolute value and the
sum of the number and it’s opposite is zero.

You have $4 and you need to pay a friend $3. What will you have after paying your friend?
4 + (-3) = 1 or (-3) + 4 = 1
7.NS.A.2
Vocab – Integers, Fractions, whole numbers
Multiplication and division of integers is an extension of multiplication and division of whole numbers.
Examples:
 Examine the family of equations. What patterns do you see? Create a model and context for each of the products.
Unit 2
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7.NS.A.3
Vocab – order of operations
Students use order of operations from 6th grade to write and solve problem with all rational numbers.
Examples:
 Your cell phone bill is automatically deducting $32 from your bank account every month. How much will the deductions total for the year?
-32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 = 12 (-32)

It took a submarine 20 seconds to drop to 100 feet below sea level from the surface. What was the rate of the descent?
Stage 2 - Evidence
Evaluative Criteria/Assessment Level Descriptors (ALDs):
7.NS.A – (SBAC Target B)
Level 4 Students should be able to apply previous understandings of operations to solve real-world problems involving rational numbers with addition,
multiplication, subtraction, and division.
Level 3 Students should be able to solve mathematical problems using the four operations on rational numbers and convert from a fraction to a decimal.
They should be able to extend previous understandings of subtraction to realize it is the same as adding the additive inverse. They should also be able to
understand p + q as a number located |q| units from p on a number line in either direction depending on the sign of q. They should also know, understand,
and use the rules for multiplying and dividing signed numbers.
Level 2 Students should be able to apply and extend previous understandings and properties of addition and subtraction to add and subtract with rational
numbers; identify the absolute value of a rational number and understand when opposites combine to make 0; and convert between familiar fractions and
decimals.
Level 1 Students should be able to add, subtract, multiply, and divide nonnegative rational numbers. They should be able to add, subtract, multiply, and
divide rational numbers with a number line or other manipulative.
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SBA Examples
Item Specs
Stage 3 – Learning Plan Sample
Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning
LEARNING ACTIVITIES:
NOTES:
**Days may change depending on any tasks or assessing you choose to do.
7.NS.A.1 & 7.NS.A.2
7.NS.A.1 & 7.NS.A.2
Day 1: Integers and Absolute value REVIEW
 Holt Course 3 Lesson 1-3
Day 2: Adding Integers (1)
 Holt Course 3 Lesson 1-4
Day 3: Subtracting Integers (1)
 Holt Course 3 Lesson 1-5
Day 4: Multiplying and Dividing Integers (1)
 Holt Course 3 Lesson 1-6
Day 5: Rational Numbers (1)
 Holt Course 3 Lesson 2-1
Day 6: Comparing & Ordering Rational Numbers (1)
 Holt Course 3 Lesson 2-2
Day 7: Adding and Subtracting Rational Numbers (1)
 Holt Course 3 Lesson 2-3
Day 8: Multiplying Rational Numbers (1)
 Holt Course 3 Lesson 2-4
Unit 2
IXL
7.NS.A.1
B.1 B.2 B.4 B.6 C.1 C.3 C.4 C.5 D.3 H.3 H.6 C.9 E.1 E.11 G.1 G.3 H.7
S.1
7.NS.A.2
A.3 A.4 A.10 C.6 C.7 C.8 C.9 E.3 E.5 E.6 E.11 F.3 G.7 G.9 G.10 G.12
G.13 G.14 H.2 H.8 H.9 S.1 S.2
Intervention
Holt Course 2 Lesson 2-1 Integers (Absolute value)
Holt Course 2 Lesson 2-2 Adding Integers (1)
Holt Course 2 Lesson 2-3 Subtracting Integers
Comparing and Ordering Rational Numbers
Holt Course 2 2-4 Multiplying and Dividing Integers
Holt Course 2 Lesson 2-10 Equivalent Fractions & Decimals
Holt Course 2 Lesson 2-11 Comparing & Ordering Rational Numbers
Performance Tasks
 What’s Your Sign? 7.NS.A.1 (MP 4)
 And the Winner IS? 7.NS.A.1
 Multiplying Integers 7.NS.A.2
 Patterns of Multiplication and Division 7.NS.A.2
 The Repeater versus the Terminator 7.NS.A.2
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Stage 3 – Learning Plan Sample
Day 9: Dividing Rational Numbers (1)
 Holt Course 3 Lesson 2-5
7.NS.A.3
IXL
C.3 C.4 C.5 C.6 C.7 C.8 E.1 E.2 E.3 E.4 E.5 E.6 E.8 G.1 G.2 G.3 G.4 G.5
G.7 G.9 G.10 G.11 G.12 G.13 G.14 G.16 H.6 H.8 M.1 M.2
7.NS.A.3
Day 10-11: Performance Task – See choices above
Common Assessment
Daily Lesson Plan
Learning Target:
Opening Activity:
Activities:
 Whole Group:
 Small Group/Guided/Collaborative/Independent:
 Whole Group:
Checking for Understanding (before, during and after):
Assessments:
Choose from the performance tasks below – students will be
solving problems with all rational numbers
Performance Tasks
 Helicopters and Submarines 7.NS.A.1/7.NS.A.3
 Debits and Credits 7.NS.A.1/7.NS.A.3
 Multiplying Rational Numbers 7.NS.A.2/7.NS.A.3
 Triple Triumph 7.NS.A.3
 Sharing Prize Money 7.NS.A.3
Comments on Guide-Please be specific:
Unit 2
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Stage 3 – Learning Plan Sample
Unit 2
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