## Unit Overview

**Essential Questions: **

**Lesson Plans and Seeds**

Lesson Seed B.4: Modeling Linear Relationships

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

**Content Emphasis By Clusters in Grade 8**

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

Lesson seeds are ideas that can be used to build a lesson aligned to the CCSS. Lesson seeds are not meant to be all-inclusive, nor are they substitutes for instruction. When developing lessons from these seeds, teachers must consider the needs of all learners. It is also important to build checkpoints into the lessons where appropriate formative assessment will inform a teachers instructional pacing and delivery.

### Unit Overview

This unit builds on prior experience with linear expressions. The focus shifts to constructing a function to model a linear relationship between two quantities, to qualitatively describe a functional relationship by analyzing a graph and conversely to sketch a graph based on the description of a relationship.

**Teacher Notes:**

- Students should be well-grounded in their knowledge of linear expressions and the properties of operations using rational numbers.
- Students should be knowledgeable about graphing in all four quadrants of the coordinate plane.
- Models should be used to demonstrating the properties of functions (i.e., constant rate of change/slope, increase/decrease, y-intercept, x-intercept, etc.).

**Enduring Understandings:**

At the completion of the unit, students will understand that:

- Functions represent rules that assign exactly one output (y-coordinate) to each input (x-coordinate) in any functional relationship.
- A variety of phenomena and relationships in mathematics can be represented, modeled, and analyzed using patterns and functions.
- Linear functions can be used to model constant rates of change.
- Not all functional relationships are linear; properties of functions can be adjusted to represent a variety of different functional relationships.
- Functions and patterns of change can be represented by using tables, graphs, words, and symbolic expressions.
- Different forms of representations have strengths and weaknesses.
- Relationships among quantities can often be expressed symbolically in more than one way.

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

- PARCC has not provided examples of opportunities for in-depth focus related to “Defining, Evaluating and Comparing Functions.”

**Possible Student Outcomes:**

Students will:

- Identify functions as linear or nonlinear and contrast their properties from tables, graphs or equations; analyze and understand different relationships between two variables.
- Determine a function to model a specific linear relationship, including its rate of change.
- Represent, analyze and generalize a variety of patterns with tables, graphs, words, and symbolic rules.
- Analyze a graph and sketch a graph, based on a specific linear relationship.
- Use precise and accurate vocabulary to compare and discuss the properties of functions.
- Use rational numbers to evaluate functions.

**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 the meaning of “domain” and “range” of a function.
- Have trouble remembering that each input (x-value) in a function can have only one output (y-value).
- Believe that all relationships having an input and an output are functions, and therefore, misuse the function terminology
- Believe that the notation
*f(x)*means to multiply some value*f*times another value*x*. The notation alone can be confusing and needs careful development. For example,*f(2)*means the output value of the function*f*when theinput value is 2. - Believe that it is reasonable to input any
*x*-value into a function, so they will need to examine multiple situations in which there are various limitations to the domains. - Believe that the slope of a linear function is merely a number used to sketch the graph of the line. In reality, slopes have real-world meaning, and the idea of a rate of change is fundamental to understanding major concepts from geometry to calculus.
- Confuse a recursive rule with an explicit formula for a function. for example, after identifying that a linear function shows an increase of 2 in the values of the output for every change of 1 in the input, some students will represent the equation as
*y=x + 2*instead of realizing that this means*y=2x + b*. - Have difficulty identifying the pattern and calculating the slope when input values are not increasing consecutive integers (e.g., when the input values are decreasing, when some integers are skipped, or when some input values are not integers).
- Not pay attention to the scale on a graph, assuming that the scale units are always one.
- Infer a cause and effect relationship between independent and dependent variables, but this is often not the case.
- Graph incorrectly because they don‘t understand that x usually represents the independent variable and y usually represents the dependent variable.

**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*

*Formative Assessments*

**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
Define, Evaluate, and Compare Functions**

** function: **A function is an association of exactly one value or object from one set of values or objects (the range) with each value or object from another set (the domain). The equation y = 3x + 5 defines y as a function of x, with the domain (x values) specified as the set of all real numbers. Thus, when y is a function of x, a value of y (the range) is associated with each real-number value of x by multiplying it by 3 and adding 5. The range is the possible values of the output, or dependent variable (y, for example). The domain is the set of possible value for the input, or the independent variable (x, for example).

** graph of a function: **The graph of a function is the set of ordered pairs (x, y) for each input value (x) and its corresponding output value (y). For the equation y = 3x + 5, ordered pairs such as (-5, -10), (-2, -1), (0, 5), (3, 14), and (300, 305), among an infinite number of addition ordered pairs.

** functional relationship: **A functional relationship describes an association of exactly one value or object from one set of values or objects (the range) with each value or object from another set (the domain).

** constant rate of change/slope: **Anything that goes up by x number of units for each y value every time is a constant rate of change. A constant rate of change increases or decreases by the same amount for every trial. The slope of a line has a constant rate of change.

** function notation: **The expression "f(x)" means "plug a value for x into a formula f "; the expression does not mean "multiply f and x." In function notation, the "x" in "f(x)" is called "the argument of the function", or just "the argument". So if we are given "f(2)" the "argument" is "2".

We evaluate "f(x)" just as we would evaluate "y".

** y-intercept: : **The point at which a line crosses the y-axis (or the y-coordinate of that point) is a y-intercept.

** linear function: **A linear function is a first-degree polynomial function of one
variable.
This type of function is known as "linear" because it includes the functions whose graph on the coordinate plane is a straight line. A linear function can be written as

*x → ax + b*or

*f(x)=mx + b*

** non-linear function:**Equations whose graphs are not straight lines are called nonlinear functions. Some nonlinear functions have specific names. A “quadratic function” is nonlinear and has an equation in the form of
, where

*a ≠ 0*. Another nonlinear function is a “cubic function”. A cubic function has an equation in the form of + cx + d, where a ≠ 0.

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

** distributive property:** Examples are a(b + c) = ab + ac and
3x(x+4) = 3x(x) + 3x(4)

** collecting like terms: **This process is also known as “combining” like terms.

**Numbers that can be expressed as an integer, as a quotient of integers (such as 1/2, 4/3 , 7), or as a decimal where the decimal part is either finite or repeats infinitely (such as 2.75 and 33.3333…) are considered rational numbers.**

*rational numbers:***Numbers that cannot be expressed as an integer, as a quotient of integers (such as 1/2, 4/3, 7), or as a decimal where the decimal part is either finite or repeats infinitely (such as 2.75 and 33.3333…) are considered irrational numbers. The values π and √2 are irrational because their values cannot be written as either a finite decimal or a repeating decimal.**

*irrational numbers:*
** volume of cones: **The formula for the volume of a cone can be determined from the volume formula for a cylinder. We must start with a cylinder and a cone that have equal heights and radii, as in the diagram below.

Imagine copying the cone so that we had three congruent cones, all having the same height and radii of a cylinder. Next, we could fill the cones with water. As our last step in this demonstration, we could then dump the water from the cones into the cylinder. If such an experiment were to be performed, we would find that the water level of the cylinder would perfectly fill the cylinder.

This means it takes the volume of three cones to equal one cylinder. Looking at this in reverse, each cone is one-third the volume of a cylinder. Since a cylinder's volume formula is V = Bh, then the volume of a cone is one-third that formula, or V = Bh/3. Specifically, the cylinder's volume formula is and the cone's volume formula is .

** volume of cylinders: **The process for understanding and calculating the volume of cylinders is identical to that of prisms, even though cylinders are curved. Here is a general cylinder.

**General Cylinder**

Here is a specific cylinder with a radius 3 units and height 4 units.

**Specific Cylinder**

We fill the bottom of the cylinder with unit cubes. This means the bottom of the prism will act as a container and will hold as many cubes as possible without stacking them on top of each other. This is what it would look like.

The diagram above is strange looking because we are trying to stack cubes within a curved space. Some cubes have to be shaved so as to allow them to fit inside. Also, the cubes do not yet represent the total volume. It only represents a partial volume, but we need to count these cubes to arrive at the total volume. To count these full and partial cubes, we will use the formula for the area of a circle.

The radius of the circular base (bottom) is 3 units and the formula for area of a circle is A = . So, the number of cubes is (3.14)(3) , which to the nearest tenth, is equal to 28.3.

If we imagine the cylinder like a building (like we did for prisms above), we could stack cubes on top of each other until the cylinder is completely filled. it would be filled so that all cubes are touching each other such that no space existed between cubes. it would look like this.

To count all the cubes above, we will use the consistency of the solid to our advantage. We already know there are 28.3 cubes on the bottom level and all levels contain the exact number of cubes. Therefore, we need only take that bottom total of 28.3 and multiply it by 4 because there are four levels to the cylinder. 28.3 x 4 = 113.2 total cubes to our original cylinder.

If we review our calculations, we find that the total bottom layer of cubes was found by using the area of a circle, . Then, we took the result and multiplied it by the cylinder's height. So the volume of a cylinder is π times the square of its radius times its height. Sometimes geometers refer to the volume as the area of the bottom times the height. Symbolically, it is written as V = Bh, where the capitol B stands for the area of the solid's base (bottom).

** volume of spheres: **A sphere is the locus of all points in a region that are equidistant from a point. The two-dimensional rendition of the solid is represented below.

**General Sphere**

To calculate the surface area of a sphere, we must imagine the sphere as an infinite number of pyramids whose bases rest on the surface of the sphere and extend to the sphere's center. Therefore, the radius of the sphere would be the height of each pyramid. One such pyramid is depicted below.

The volume of the sphere would then be the sum of the volumes of all the pyramids. To calculate this, we would use the formula for volume of a pyramid, namely V = Bh/3. we would take the sum of all the pyramid bases, multiply by their height, and divide by 3.