## Unit Overview

**Essential Questions: **

**Lesson Plans and Seeds**

Lesson Plan C.6-7: Graphing Coordinates

Lesson Seed C.5: Introduction of Integers

**Download Seeds, Plans, and Resources (zip)**

**Content Emphasis By Clusters in Grade 6**

**Progressions from Common Core State Standards in Mathematics**

### Unit Overview

This is the first time that students have formally studied rational numbers. The study of rational numbers extends students’ work from whole numbers, fractions and decimals in elementary grades. This unit mostly deals with integers. Integers are the foundation for further study in mathematics, science and are useful in everyday life. Students will be working with rational numbers to graph, find reflections and distances on a coordinate plane, understand absolute value, and solve situations in daily life.

**Teacher Notes:**

- Through working with real life situations, teachers should help students see the need for positive and negative numbers..
- Teachers should prepare students to understand the definition of positive and negative numbers.
- Use graphing to show position and magnitude.
- Solve real world problems by graphing on a coordinate plane.

**Enduring Understandings:**

At the completion of this unit on applying and extending students’ previous understanding of numbers to the rational number system, the student will understand that:

- Rational numbers allow us to make sense of situations that involve numbers that are not whole.
- Rational numbers can be represented in multiple ways.
- The value of a number is determined by its position on the number line.
- Integers have magnitude and direction.

**Focus Standards (Listed as Examples of Opportunities for In-Depth Focus in the PARCC Content Framework document):
**

**6.NS.C.8**When students work with rational numbers in the coordinate plane to solve problems, they combine and consolidate elements from the other standards in this cluster.

**Possible Student Outcomes:**

The student will be able to:

- understand that positive and negative numbers are used to describe quantities having opposite directions or values.
- graph rational numbers on a number line and in a coordinate plane.
- solve authentic, mathematical problems by graphing points in all four quadrants of the coordinate plane.

**Evidence of Student Learning:**

*The Partnership for Assessment of Readiness for College and Careers (PARCC) has awarded the Dana Center a grant to develop the information for this component. This information will be provided at a later date. The Dana Center, located at the University of Texas in Austin, encourages high academic standards in mathematics by working in partnership with local, state, and national education entities. Educators at the Center collaborate with their partners to help school systems nurture students' intellectual passions. The Center advocates for every student leaving school prepared for success in postsecondary education and in the contemporary workplace.*

**Fluency Expectations and Examples of Culminating Standards:**

**Common Misconceptions:**

Students may

- confuse mathematics terms such as positive, negative, opposites, reflections, integers, rational numbers, absolute value, and magnitude.
- not understand how to set up a table or graph.
- not understand the difference between an integer and a rational number.

**Interdisciplinary Connections:**

Interdisciplinary connections fall into a number of related categories:

*Literacy standards within the Maryland Common Core State Curriculum*

*Science, Technology, Engineering, and Mathematics standards*

*Instructional connections to mathematics that will be established by local school systems, and will reflect their specific grade-level coursework in other content areas, such as English language arts, reading, science, social studies, world languages, physical education, and fine arts, among others.*

**Sample Assessment Items: ***The items included in this component will be aligned to the standards in the unit and will include:*

*Items purchased from vendors*

*PARCC prototype items*

*PARCC public release items*

*Maryland Public release items*

**Interventions/Enrichments/PD: ***(Standard-specific modules that focus on student interventions/enrichments and on professional development for teachers will be included later, as available from the vendor(s) producing the modules.)*

**Vocabulary/Terminology/Concepts: ***This section of the Unit Plan is divided into two parts. Part I contains vocabulary and terminology from standards that comprise the cluster, which is the focus of this unit plan. Part II contains vocabulary and terminology from standards outside of the focus cluster. These “outside standards” provide important instructional connections to the focus cluster.*

**
Part I – Focus Cluster
Apply and Extend Previous Understandings of Numbers to the System of Rational Numbers **

** reflections:** a mirror image that is sometimes called a flip. It is a transformation or rigid motions that do not change the size or shape of the object being moved. An example in the coordinate plane is:

** integers:** A whole number and their opposites. Integers do not include fractions or decimals. Zero is not considered to be negative or positive. The positive sign is not always written with its number but the negative sign must always be there.

Example: 2 and -2, 10 and -10, 82 and -82 -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5

** rational numbers:** A rational number is a number that can be in the form

^{p}/

_{q}where

**p**and

**q**are integers and

**q**is not equal to zero.

Example:

^{1}/

_{3}

^{-7}/

_{2}

^{4}/

_{1}

^{6}/

_{-17}

** absolute value: ** : the absolute value of a number as its distance from zero. The absolute value of x, denoted "| x |" (and which is read as "the absolute value of x"), is the distance of x from zero. This is why absolute value is never negative; absolute value only asks "how far?", not "in which direction?" This means not only that | 3 | = 3, because 3 is three units to the right of zero, but also that | –3 | = 3, because –3 is three units to the left of zero.

**Part II – Instructional Connections outside the Focus Cluster**

** ratio:** Ratio is a comparison of two quantities or measures. Ratios can be expressed in the form (

^{a}/

_{b}), a to b, or a:b.

Ratios can be expressed as comparisons of:

- part to a whole, one part of a whole to another part of the same whole. Part-to-whole would be the ratio of boys to the whole class. measures of two different types which is called a rate.
- part-to-part would be the ratio of boys to girls in a class.

Measures of two different types are called a rate. Rate would be the ratio of miles per gallon to miles per hour.

** unit rate:** A ratio where the denominator is 1 unit. Example: If 15 bars of soap cost $6.75, one bar would cost $.45.

^{6.75}/

_{15}=

^{.45}/

_{1}

** tape diagrams: ** Tape diagrams are linear models used to represent data and help students organize their thinking. Example: Casey read 7 more books than Jamie. If Casey has read 16 books, how many books did Jamie read?

** percent:** is another name for hundredths Ratio of a number to 100 with a percent sign. Per hundred

Example: 75% =

^{75}/

_{100}= .75