Essential Questions:
Lesson Plan A.1: Functions and Graphing
Lesson Seed A.1: Vertical Line Test
Unit Overview
Content Emphasis By Clusters in Grade 8
Progressions from Common Core State Standards in Mathematics
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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.
This unit builds on prior experience with linear expressions. The focus then shifts to expressing them with verbal descriptions, algebraically, graphically, and numerically in tables to develop an understanding of linear and non-linear functions.
At the completion of the unit, students will understand that:
Students will be able to:
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.
Students may
Interdisciplinary Connections:
Interdisciplinary connections fall into a number of related categories:
Sample Assessment Items: The items included in this component will be aligned to the standards in the unit and will include:
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.
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.
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