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Math for College Readiness | Unit A: Real Numbers
Unit Overview
Math Florida Standards
Content Standards
Major Focus: Students will understand the operations of real numbers and will distinguish rational and irrational
numbers.
Tasks:


Students will add, subtract, multiply, and divide rational and irrational numbers.
Students will use approximation techniques to order real numbers.
Standards for
Mathematical Practice
MAFS.7.NS.1.1
MAFS.K12.MP.2.1
MAFS.7.NS.1.2
MAFS.K12.MP.3.1
MAFS.8.NS.1.1
MAFS.K12.MP.5.1
MAFS.8.NS.1.2
MAFS.K12.MP.6.1
MAFS.912.N-RN.2.3
Textbook Resources
Martin-Gay, E. (2013). Intermediate algebra: Math for
college readiness. Boston, MA: Pearson Learning
Solutions.
Sections: 1.2, 1.3
Tips for Success in Mathematics: pp. 2-6
Study Skills Builder: pp. 663-671
Bigger Picture Study Guide Outline: pp. 672-674
Mathematics Formative Assessment System Tasks
The system includes tasks or problems that teachers
can implement with their students, and rubrics that
help the teacher interpret students' responses.
Teachers using MFAS ask students to perform
mathematical tasks, explain their reasoning, and justify
their solutions. Rubrics for interpreting and evaluating
student responses are included so that teachers can
differentiate instruction based on students' strategies
instead of relying solely on correct or incorrect
answers. The objective is to understand student
thinking so that teaching can be adapted to improve
student achievement of mathematical goals related to
the standards. Like all formative assessment, MFAS is a
process rather than a test. Research suggests that welldesigned and implemented formative assessment is an
effective strategy for enhancing student learning.
http://www.cpalms.org/resource/mfas.aspx
This a working document that will continue to be revised and improved taking your feedback into consideration.
Other Resources
Mathematics Assessment Resource Service
College Readiness Math Resources
PERT Resources
Algebra Nation
Online Graphing Calculator
National Library of Virtual Manipulatives
Geogebra
Virtual Nerd
YouTube
Khan Academy—Math
Engage NY
TI Nspired Resource Center for Educators
Pasco County Schools, 2014-2015
Math for College Readiness | Unit A: Real Numbers
Unit Scale (Multidimensional) (MDS)
The multidimensional, unit scale is a curricular organizer for PLCs to use to begin unpacking the unit. The MDS should not be used directly with students and is not for
measurement purposes. This is not a scoring rubric. Since the MDS provides a preliminary unpacking of each focus standard, it should prompt PLCs to further explore question #1,
“What do we expect all students to learn?” Notice that all standards are placed at a 3.0 on the scale, regardless of their complexity. A 4.0 extends beyond 3.0 content and helps
students to acquire deeper understanding/thinking at a higher taxonomy level than represented in the standard (3.0). It is important to note that a level 4.0 is not a goal for the
academically advanced, but rather a goal for ALL students to work toward. A 2.0 on the scale represents a “lightly” unpacked explanation of what is needed, procedural and
declarative knowledge i.e. key vocabulary, to move students towards proficiency of the standards.
4.0
In addition to displaying a 3.0 performance, the student must demonstrate in-depth inferences and applications that go beyond what was taught within these
standards. Examples:


3.0
Given a list of rational and irrational numbers.
o Order the numbers from least to greatest
o Provide an example of a rational and an irrational number that lie between each given number
Given that the addition of integers is closed under addition and multiplication, prove that rational numbers are closed under addition and multiplication.
The Student will:
 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.
b.
c.
d.

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.
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.
Understand subtraction of rational numbers as adding the 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.
Apply properties of operations as strategies to add and subtract rational numbers. (MAFS.7.NS.1.1)
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
a.
b.
c.
d.
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.
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.
Apply properties of operations as strategies to multiply and divide rational numbers.
Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
(MAFS.7.NS.1.2)
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Math for College Readiness | Unit A: Real Numbers
2.0
1.0

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show
that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. (MAFS.8.NS.1.1)

Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and
estimate the value of expressions (e.g., π²). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and
1.5, and explain how to continue on to get better approximations. (MAFS.8.NS.1.2)

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the
product of a nonzero rational number and an irrational number is irrational. (MAFS.912.N-RN.2.3)
The student will recognize or recall specific vocabulary, such as:
 Addition, Subtraction, Multiplication, Division, Sum, Difference, Product, Quotient, Rational, Irrational, Decimal Expansion, Approximation, Absolute
Value, Square Root
The student will perform basic processes, such as:
 Add and subtract rational numbers

Multiply and divide rational numbers and understand the rules that apply to positive and negative numbers when multiplying and dividing

Classify numbers as rational and irrational based on their decimal expansions

Determine decimal approximations of irrational numbers

Find the sum and product of two rational numbers and find the sum and product of a rational number and an irrational number
With help, partial success at 2.0 content but not at score 3.0 content
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Math for College Readiness | Unit A: Real Numbers
Unpacking the Standard: What do we want students to Know, Understand and Do (KUD):
The purpose of creating a Know, Understand, and Do Map (KUD) is to further the unwrapping of a standard beyond what the MDS provides and assist PLCs in answering question
#1, “What do we expect all students to learn?” It is important for PLCs to study the focus standards in the unit to ensure that all members have a mutual understanding of what
student learning will look and sound like when the standards are achieved. Additionally, collectively unwrapping the standard will help with the creation of the uni-dimensional
scale (for use with students). When creating a KUD, it is important to consider the standard under study within a K-12 progression and identify the prerequisite skills that are
essential for mastery.
Domain: Number & Quantity: The Real Number System
Cluster: Use properties of rational and irrational numbers. (Additional)
Standard: MAFS.912.N-RN.2.3: (Explain) why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational;
and the product of a nonzero rational number and an irrational number is irrational.
Understand
“Essential understandings,” or generalizations, represent ideas that are transferable to other contexts.
Students should understand that addition and multiplication of rational numbers will result in a rational number, and that the inclusion of one irrational number to those
operations with nonzero rational numbers will result in an irrational number.
Know
Declarative knowledge: Facts, vocab., information
Do
Procedural knowledge: Skills, strategies and processes that are transferrable to other contexts.
Vocabulary: rational, irrational, nonzero, sum,
product
Find the sum and product of two rational numbers and the sum and product of a rational number and an irrational
number.
Rational numbers are written in the form p/q,
where p and q are integers and q ≠ 0.
Explain why the sum and product of two rational numbers will be rational.
Explain why the sum and product of a rational number and an irrational number will be irrational.
A number cannot be both rational and irrational.
Prerequisite skills: What prior knowledge (foundational skills) do students need to have mastered to be successful with this standard?
Addition, subtraction, multiplication, and division of rational numbers; a basic understanding of roots resulting in an irrational number and π
Learning Goals:
Students can find the sum and product of two rational numbers and the sum and product of a rational number and an irrational number
Students can explain why the sum and product of two rational numbers will be rational.
Students can explain why the sum and product of a rational number and an irrational number will be irrational.
Moving Beyond:
MPRCC4—Add, subtract, multiply, and divide integers, fractions, and decimals.
MPRCC10—Use calculators appropriately and make estimations without a calculator regularly to detect potential errors.
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Math for College Readiness | Unit A: Real Numbers
Uni-Dimensional, Lesson Scale:
The uni-dimensional, lesson scale unwraps the cognitive complexity of a focus standard for the unit, using student friendly language. The purpose is to articulate distinct levels of
knowledge and skills relative to a specific topic and provide a roadmap for designing instruction that reflects a progression of learning. The sample performance scale shown
below is just one example for PLCs to use as a springboard when creating their own scales for student-owned progress monitoring. The lesson scale should prompt teams to
further explore question #2, “How will we know if and when they’ve learned it?” for each of the focus standards in the unit and make connections to Design Question 1,
“Communicating Learning Goals and Feedback” (Domain 1: Classroom Strategies and Behaviors). Keep in mind that a 3.0 on the scale indicates proficiency and includes the
actual standard. A level 4.0 extends the learning to a higher cognitive level. Like the multidimensional scale, the goal is for all students to strive for that higher cognitive level,
not just the academically advanced. A level 2.0 outlines the basic declarative and procedural knowledge that is necessary to build towards the standard.
MAFS.912.N-RN.2.3: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and the
product of a nonzero rational number and an irrational number is irrational.
Score
4.0
3.5
Learning Progression
I can…
 Prove that the sum of a rational number and an irrational number will
make an irrational number by considering the decimal expansion of
the numbers in three specific cases.
I can do everything at a 3.0, and I can demonstrate partial success at score 4.0.
I can…
 Explain why the sum and a product of two rational numbers will be
rational.
 Explain why the sum and a product of a rational number and an
irrational number will be irrational.
3.0
Sample Tasks
1. Consider the expression n + π where n is a rational number and π is the
irrational number 3.1415…. Using decimal expansion, show that n + π will
result in an irrational number when n is…
a) An integer.
b) A rational number with a terminating decimal.
c) A rational number with a repeating decimal.
1. a) Consider the sum
1 æ 3ö
+ ç - ÷. Will the result be rational or irrational?
4 è 5ø
b) Explain your answer from a).
c) Prove your classification by computing the sum.
2. a) Consider the product 3× 2 . Will the result be rational or irrational?
b) Explain your answer from a).
c) Prove your classification by computing the product. (
2.5
2.0
1.0
I can do everything at a 2.0, and I can demonstrate partial success at score 3.0.
I can…
 Find the sum and product of rational numbers
 Find the sum and product of a rational number and an irrational
number
I need prompting and/or support to complete 2.0 tasks.
1. Add:
2 =1.414...)
1 æ 3ö
+ ç- ÷
4 è 5ø
2. Using decimal expansion, approximate the value of
This a working document that will continue to be revised and improved taking your feedback into consideration.
3 2 . ( 2 =1.414...)
Pasco County Schools, 2014-2015
Math for College Readiness | Unit A: Real Numbers
Sample High Cognitive Demand Tasks:
These task/guiding questions are intended to serve as a starting point, not an exhaustive list, for the PLC and are not intended to be prescriptive. Tasks/guiding questions simply
demonstrate one way to help students learn the skills described in the standards. Teachers can select from among them, modify them to meet their students’ needs, or use them
as an inspiration for making their own. They are designed to generate evidence of student understanding and give teachers ideas for developing their own activities/tasks and
common formative assessments. These guiding questions should prompt the PLC to begin to explore question #3, “How will we design learning experiences for our students?”
and make connections to Marzano’s Design Question 2, “Helping Students Interact with New Knowledge”, Design Question 3, “Helping Students Practice and Deepen New
Knowledge”, and Design Question 4, “Helping Students Generate and Test Hypotheses” (Domain 1: Classroom Strategies and Behaviors).
MAFS Mathematical Content Standard
Design Question 1; Element 1
MAFS Mathematical Practice(s)
Design Question 1; Element 1
Marzano’s Taxonomy
Teacher Notes
MAFS.912.N-RN.2.3: Explain why the sum or product of two rational numbers is rational; that the sum of a
rational number and an irrational number is irrational; and the product of a nonzero rational number and an
irrational number is irrational.
MAFS.K12.MP.2.1: Reason abstractly and quantitatively.
MAFS.K12.MP.3.1: Construct viable arguments and critique the reasoning of others.
MAFS.K12.MP.5.1: Use appropriate tools strategically.
MAFS.K12.MP.6.1: Attend to precision.
Level 3-Analysis: “Classifying”
Questions:
Why are your answers different?
Can you defend your answer with mathematical reasoning?
What is the root word of “rational”?
Misconceptions:
Incorrect computations (fraction operation, decimal approximation)
Truncating decimals of radical approximations
Differentiation:
More complicated examples for extending students
Simplified examples for struggling students
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Math for College Readiness | Unit A: Real Numbers
1. Perform the following computations:
a.
Task
6 4
×
7 3
b.
3 + 4 ( 3 =1.732...)
2. Classify your results as rational or irrational.
3. Explain what caused the difference in classification with your results.
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015