## Unit Overview

**Essential Questions: **

**Lesson Plans and Seeds**

Lesson Plan B.5: Graphing Proportional Relationships

Lesson Seed B.5: Families of Graphs

Lesson Seed B.6: Similar Right Triangles and Slope

**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 plan extends knowledge of ratios and proportional relationships from grade 6 and grade 7 to linear functions in grade 8. This plan focuses on: interpreting unit rate as a constant rate of change; interpreting m in the equations y = mx and y = mx + b as a constant rate of change/slope; and comparing different proportional relationships presented in tables, graphs, and equations.

**Teacher Notes:**

- Students should have experience with proportional relationships in the context of scale drawings and for making inferences about a given population from a random sample.
- Models should be used to demonstrate the properties of constant rate of change/slope.
- Students should understand the usefulness of technology, such as the graphing calculator, to experiment with proportional relationships, lines, and linear equations.

**Enduring Understandings:**

At the completion of the unit on Connections between Proportional Relationships, Lines, and Linear Equations the student will understand that:

- The unit rate for a data set that represents a proportional relationship can be interpreted as slope when the data is graphed on a coordinate plane.
- The slope m is the same for any two distinct points on a non-vertical line graphed on the coordinate plane.
- The formula y = mx is another way of expressing direct variation y = kx; both m and k represent constant values in a proportional relationship.
- Graphs of linear equations that intersect the y-axis at any point other than the origin (0, 0) do not represent proportional relationships.

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

**8.EE.B.5**When students work toward meeting this standard, they build on grades 6-7 work with proportions. In doing so, students position themselves for grade 8 work with functions and the equation of a line.

**Possible Student Outcomes:**

Students will be able to:

- Interpret unit rate as the slope of a linear equation to graph proportional relationships on the coordinate plane.
- Recognize and apply direct variation to understand that all proportional relationships are linear in the form y = mx. When graphed, this line intersects the origin. In an authentic scenario, the graph of a direct variation tends to be in Quadrant I.
- Recognize and define slope to differentiate between linear equations with positive, negative, undefined, or zero slope.
- Analyze different proportional relationships to compare related data from tables, graphs, or linear equations.

**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

- Neglect to state directionality (left/right or positive/negative) on the x-axis when describing ∆x; simply say “over,” which is too vague
- Confusing with the ordered pair notation (x, y)
- Mix the meanings of x (independent variable) and y (dependent variable), particularly when graphing the line of an equation
- Confuse a horizontal line (slope of zero) with a vertical line (undefined slope).
- Mistakenly believe that a slope of zero is the same as “no slope.”

**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
Connections between Proportional Relationships, Lines, and Linear Relationships**

**Slope of a line is a ratio that describes the steepness of a line. Slope is usually written in fraction form that places the value of the horizontal change along the x-axis in the denominator of the fraction and the vertical change along the y-axis in the numerator. In the diagram, the horizontal change (run) is 2, while the vertical change (rise)**

*Slope:*is 3. Thus, the slope of the line is 3/2

**Unit rate is the ratio of two different measurements in which the second term is 1. Example: 6:1, 6 miles/one gallon, (In fraction form, the denominator is 1.)**

*Unit rate:*
** Proportional relationship: **A comparison of two variable quantities having a fixed (constant) ratio is considered to be a proportional relationship. Example: 50 miles on 3 gallons is a proportional relationship to 100 miles on 6 gallons and 150 miles on 9 gallons.

** chord: **A chord is a line segment that has both endpoints on the circumference of a circle.

*Constant rate of change/slope:*The line graphed on this coordinate plane has a slope of 3/2. This means as we move up or down the line, the ratio of the change in units on the x-axis, 2, is constant with the change in units on the y-axis, 3.

slope =

*zero slope:*As shown in the diagram, when the slope of a line is zero, the line is horizontal and parallels the x-axis. As we move left or right on the line, the value for y does not change. In the diagram, the y-value is always 5, regardless of the x-value.

If we think of y/x as , then zero divided by any value is always zero.

For example, if we move on the line from the point (-4, 5) to the point (3, 5),

then y/x = 0/7 = 0. Conversely, if you move from (3, 5) to (-4, 5), then y/x = 0/-7 = 0.

*undefined slope:*As shown in the diagram, when the slope of a line is undefined, the line is vertical and parallels the y-axis. As one moves up or down a line, the value for x does not change. The x-value for the line on the right is always 3, regardless of the y-value.

We know that division by zero is impossible because regardless of the value of the dividend, it can never be reproduced through multiplication by a divisor of zero because 0 x 0 = 0.

So, if we think of x/y as , then the slope of a vertical line is incapable of being defined.

For example, if you move on the line from the point (3, 7) to the point (3, 2), then y/x = -5/0 = undefined. Conversely, if you move from (3, 2) to (3, 7), then x/x = 5/0 = undefined.

** families of graphs: ** The graph below shows a few family members for the linear equation y = x.

The line that intersects the origin (0, 0), y = x, is considered the “parent.” All members of a family of graphs have the same slope, but different y-intercepts. Families of graphs can have infinitely many members. |

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

** one solution: **An example of a linear equation in one variable with one solution is x = 5.

** infinitely many solutions: **An example of a linear equation in one variable with infinitely many solutions is x = x.

** no solutions: **An example of a linear equation in one variable with no solutions is x + 1 = x + 2.

** equivalent equation: **: Equations with the same value are equivalent, for example: 2 + 3 = 11 − 6 is equivalent to 5 = 5 OR y +3 = 5 is equivalent to y = 2.

** distributive property: **The distributive property states that for all numbers a, b, and c, a(b+c) = ab + ac. 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.

Example: 2x + 3 + 5x = 7x +3

** 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.

** 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

** transformations: **anytime you move, shrink, or enlarge a figure, you have to make a transformation of that figure. This kind of transformation includes rotations, reflections, translations and dilations.

Rotation |
"Turn" |
||

Reflection |
"Flip" |
||

Translation |
"Slide" |
||

Dilation |
"Resizing" |

** rotations: **It is also called a turn. Rotating a figure means turning the figure around a point. The point can be on the figure or it can be some other point. The shape still has the same size, area, angles and line lengths.

** reflections: **It is also called a flip. When a figure is flipped over a line. Each point in a reflection image is the same distance from the line as the corresponding point in the original shape. The shape still has the same size, area, angles and line lengths.

** translations: **It is also called a slide. In a translation, every point in the figure slides the same distance in the same direction. The shape still has the same size, area, angles and line lengths.

** dilations: **It is also called resizing. The shape becomes bigger or smaller. If you have to dilate or resizing to make one shape become another, they are similar. These figures are non-rigid because they do not stay the same.

** scale factor: **The amount by which the image grows (dilates) or shrinks (reduces) is called the "Scale Factor". The general formula for dilating a point with coordinates of (2, 4) by a scale factor of ½ is: (2 • ½, 4 • ½) or (1, 2); scale factor of 1 produces a congruent figure: (2 • 1, 4 • 1) or (2, 4).

*transformation notation: *

** Preimage** is the figure prior to the transformation. (A, B, C)

**is the figure after the transformation. (A', B', C') A', B', C ' are called A prime, B prime, and C prime. A A' B B' C C'**

*Image*

**https://learnzillion.com/lesson_plans/8062-find-a-unit-rate-using-a-graph****http://baltimore.cbslocal.com/2011/09/04/eastern-shore-shows-its-crab-picking-might-at-crab-derby/**- Geometer’s Sketchpad
- Geometer’s Sketchpad 8.EE.6 Lesson Seed.gsp
**http://commoncoretools.files.wordpress.com/2011/04/ccss_progression**

_ee_2011_04_25.pdf