Algebra I, Quarter 1, Unit 1.1 Reasoning Quantitatively

Algebra I, Quarter 1, Unit 1.1

Reasoning Quantitatively

Overview

Number of instruction days: 6–8 (1 day = 53 minutes)

Content to Be Learned Mathematical Practices to Be Integrated

Determine and interpret scales on graphs. 2 Reason abstractly and quantitatively.
Select the necessary information for problem
Define quantities for modeling problems. solving.

Choose appropriate level of accuracy of 4 Model with mathematics.
measurement. Interpret scale and origin on graphs.
Use appropriate quantities in displaying
Explain that the sum or product of rational descriptive modeling.
numbers is a closed set. Show limitations on measurements when
working with quantities.
Explain that the sum of an irrational number
and a rational number is irrational. 6 Attend to precision.
Use correct units, signs, labels, and write
Explain that the product of a nonzero rational answers in sentence form.
number and an irrational number is irrational. Use appropriate measurement for accuracy.

Essential Questions How do you determine the appropriate level of
accuracy for a quantitative solution?
What is the relationship between irrational and
rational numbers within the real number What type of real numbers would a student find
system? in applying the arithmetic operations of
addition and multiplication to any combination
How do you determine the appropriate of rational and irrational numbers?
graphical display for a given problem?

Why are units important in the problem-solving
process?

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Algebra I, Quarter 1, Unit 1.1 Reasoning Quantitatively (6–8 days)
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Standards

Common Core State Standards for Mathematical Content

Number and Quantity

Quantities★ N-Q

Reason quantitatively and use units to solve problems.

N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems;
choose and interpret units consistently in formulas; choose and interpret the scale and the
origin in graphs and data displays.★

N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.★

N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting
quantities.★

The Real Number System N-RN

Use properties of rational and irrational numbers.

N-RN.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 that the product of a nonzero rational
number and an irrational number is irrational.

Common Core State Standards for Mathematical Practice

2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations.
They bring two complementary abilities to bear on problems involving quantitative relationships: the
ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the
representing symbols as if they have a life of their own, without necessarily attending to their referents—
and the ability to contextualize, to pause as needed during the manipulation process in order to probe into
the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent
representation of the problem at hand; considering the units involved; attending to the meaning of
quantities, not just how to compute them; and knowing and flexibly using different properties of
operations and objects.

4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in
everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition
equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a
school event or analyze a problem in the community. By high school, a student might use geometry to
solve a design problem or use a function to describe how one quantity of interest depends on another.
Mathematically proficient students who can apply what they know are comfortable making assumptions
and approximations to simplify a complicated situation, realizing that these may need revision later. They
are able to identify important quantities in a practical situation and map their relationships using such

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Reasoning Quantitatively (6–8 days) Algebra I, Quarter 1, Unit 1.1
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tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret their mathematical results in the context of
the situation and reflect on whether the results make sense, possibly improving the model if it has not
served its purpose.

6 Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the symbols
they choose, including using the equal sign consistently and appropriately. They are careful about
specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.
They calculate accurately and efficiently, express numerical answers with a degree of precision
appropriate for the problem context. In the elementary grades, students give carefully formulated
explanations to each other. By the time they reach high school they have learned to examine claims and
make explicit use of definitions.

Clarifying the Standards

Prior Learning

In Grade 6, students were introduced to systems of rational numbers. They were responsible for becoming
fluent in multiplying and dividing rational numbers. They analyzed the relationship between independent
and dependent variables. In Grade 7, students applied and expanded previous understanding of rational
numbers (in real-world settings and/or with equations). In Grade 8, students were introduced to irrational
numbers. They converted rational numbers to show their decimal form and approximated irrational
numbers.

Current Learning

Algebra I students extend their knowledge of rational and irrational numbers to explain why the sum or
product of two rational or irrational numbers is rational or irrational. Students learn how to interpret units,
choose appropriate scales, and create the coordinate system for graphs and/or data displays. These skills
will be revisited in more detail throughout the course. Students also define the meaning of variables in the
context of given problems. They are exposed to determining appropriate limitations on measurements
when reporting quantities (knowing when to round up or down within the context of a given problem).

Future Learning

Algebra II students will perform arithmetic operations involving complex numbers. They will also use
complex numbers in polynomial identities and equations. Students in fourth-year courses will represent
and model quantities with vectors and matrices. Rational and irrational numbers will be used in all future
courses.

Additional Findings

One challenge that students face when studying graphs is that “the complexity that teachers and
researchers now state in function graphs flows from the fact that a graph has many potential meanings and
can be interpreted in many different ways.” (A Research Companion to Principles and Standards, p. 250)

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Students may also have difficulty in building a problem model: “In building a problem model, students
need to be alert to the quantities in the problem. It is particularly important that students represent the
quantities mentally, distinguishing what is known from what is to be found.” (Adding It Up, p.125)

It is hard for students to learn about rational numbers. “Learning about rational numbers is more
complicated and difficult than learning about whole numbers. Rational numbers are more complex than
whole numbers in part because they are represented in several ways.” (Adding It Up, p.231)

Assessment

When constructing an end-of-unit assessment, be aware that the assessment should measure your
students’ understanding of the big ideas indicated within the standards. The CCSS for Mathematical
Content and the CCSS for Mathematical Practice should be considered when designing assessments.
Standards-based mathematics assessment items should vary in difficulty, content, and type. The
assessment should comprise a mix of items, which could include multiple choice items, short and
extended response items, and performance-based tasks. When creating your assessment, you should be
mindful when an item could be differentiated to address the needs of students in your class.

The mathematical concepts below are not a prioritized list of assessment items, and your assessment is
not limited to these concepts. However, care should be given to assess the skills the students have
developed within this unit. The assessment should provide you with credible evidence as to your students’
attainment of the mathematics within the unit.

Determine and interpret scales on a graph

Use units of measure in real world problems

Estimate rational and irrational numbers with appropriate levels of accuracy

Classify and use real numbers in real world problems

Add, subtract, multiply and divide with rational and irrational numbers

Identify sums and products in the real number system

Instruction

Learning Objectives

Students will be able to:
Determine and interpret scales on a graph
Define quantities for modeling problems
Choose appropriate levels of accuracy of measurement
Explain that the sum or product of rational numbers is a closed set
Explain that the sum of an irrational number and a rational number is irrational
Explain that the product of a nonzero rational and an irrational number is irrational
Demonstrate understanding of concepts and skills learned in this unit.

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Reasoning Quantitatively (6–8 days) Algebra I, Quarter 1, Unit 1.1
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Resources

Algebra 1, (Glencoe McGraw Hill) 2010
Section 0-1 pp. P5 to P6
Section 0-2 pp. P7 to P10
Section 0-3 pp. P11 to P12
Section 0-4 pp. P13 to P16
Section 0-5 pp. P17 to P19
Section 0-13 pp. P40 to P43
Student Handbook pp. 804 to 814

Quick Review Math Handbook, (Glencoe McGraw Hill) 2010
Sections 2, 3.1, 3.2, 4.2, 4.3 (Select problems based on Standards for this unit)
Interactive Classroom CD
Teacher Works Plus CD-ROM
Teaching with Foldables (Dinah Zike; Glencoe McGraw Hill 2010)

Exam View Assessment Suite
Math Online, glencoe.com
The following websites offer representations of real-world data and can be used with students to interpret
scales and define quantities.

www.usatoday.com/snapshot/news/snapndex.htm
www.newsingraphs.com/
www.econoclass.com/misleadingstats.html

Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the
Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations.

Materials

Ti-Nspire calculators

Instructional Considerations

Key Vocabulary
No new vocabulary for this unit.

Planning for Effective Instructional Design and Delivery

The reinforced vocabulary taught in previous grades or units includes variable, integers, rational
numbers, irrational numbers, absolute value, reciprocal, frequency table, bar graph, line graph,
histogram, and stem and leaf.

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Algebra I, Quarter 1, Unit 1.1 Reasoning Quantitatively (6–8 days)
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The focus of this unit is number sense. The content in this unit draws from student’s work in previous
grades. It serves to help students recall prior knowledge and refresh their skills which will help them be
successful in their future work in algebra. As students work through this unit they may find it helpful to
organize their learning through a study organizer such as a foldable (Dinah Zike’s Teaching with

Foldables). Consider using the layered-look book (p. 10). It will help organize the material by content and
students can use it as a study guide for the unit assessment. It is not possible to have students work on all
problems in each of the sections listed above. Choose items that closely align to the standards.

To review real numbers have students create a nonlinguistic representation such as a Venn diagram to
organize and classify all numbers. Remind students to be consistent with scale marks on number lines and
graphs. Students should use tick marks to show equal intervals. As a formative assessment students can
write down a rational and irrational number and label each one as rational and irrational. Have them turn
it in as an exit ticket. The same strategy can be used for sums and products of rational and irrational
numbers. For example, students can write out a problem of adding a rational and irrational number to
show that the sum is irrational.

Reviewing quantities and graphs is a good opportunity for students to work with real world contexts. The
following websites offer many representations of real world data and provide the students with the
opportunity to interpret scales and define quantities.

www.usatoday.com/snapshot/news/snapndex.htm

www.newsingraphs.com/

www.econoclass.com/misleadingstats.html

Incorporate the essential questions as part of the daily lesson. Options include using them as a “do now”
to activate prior knowledge of the previous day’s lesson, using them as an exit ticket by having students
respond to it and post it, or hand it in as they exit the classroom, or using them as other formative
assessments. Essential questions could also be used in the unit assessment.

For planning considerations read through the teacher edition for suggestions about scaffolding techniques,
using additional examples, and differentiated instructional guidelines as suggested by the Glencoe
resource.

When working with order of operations, you may use the TI-Nspire Calculator so that students can
become familiar with the key locations.

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Reasoning Quantitatively (6–8 days) Algebra I, Quarter 1, Unit 1.1
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Notes

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