9.2.1B Parabolas, Asymptotes & Graphs
Overview
Standard 9.2.1 Essential Understandings
The concept of function is one of the most important ideas in school mathematics. Functions are a way to study the relationship among two or more variables and organize the world. The definition of a mathematical function developed over a thousand years. The beginning focus was on the covariation of two variables. Studying how two variables change together constitutes much of the middle and secondary curriculum. As the definition of function evolved the "exactly one" feature of the function gained importance. This feature is highlighted when students are asked to distinguish between relations that are functions and those that are not.
In elementary school, students begin to study functions by finding rules for various patterns. The rule for determining the number of toothpicks in a figure is a function relating the figure number (independent variable) with the number of toothpicks in that figure (dependent variable).
| |||
figure number | 1 | 2 | 3 |
number of toothpicks in figure | 5 | 10 | 15 |
Students explain that the number of toothpicks is the figure number multiplied by five.
Thinking of functions as rules that generalize patterns is developed further in middle school where students learn that a proportional relationship between two variables exists if the rule relating the variables is written as . They also can see this relationship graphically as a line passing through the origin. At the end of middle school, students are able to distinguish between linear functions and functions that are not linear by inspecting the function represented by a table, graph, equation, or a real-world situation.
Students at the secondary level use their knowledge of how linear functions change and relate it to how exponential and quadratic functions change. Among the functions that students study in high school is the rate of change as shown in tables, graphs, and equations, one of the most important features that distinguish the different types. Parts of the curves of quadratic and exponential functions can appear to be similar on a graph, but the differences in the rates of change show that no quadratic or exponential will ever match at more than just a few discrete points. Quadratic functions and exponential functions can both increase at an increasing rate but these patterns of change are quite different.
Functions are tools that people use to model relationships between two variables. In middle school students use linear functions to model real-world situations. The function w(t) = 1.5t + 7 could represent to weights of baby girls when they are t months old. Students learn how special features of the function relate to the real-world situation. The y-intercept represents the weight of the baby girl at birth, while the slope represents the number of pounds that the baby gains each month. In addition, the domain of the function is determined by the constraints of the real-world setting. Theoretically, the domain of any linear function could be the set of real numbers, but practically the function does not really make sense for values of t that are less than 0 months or larger than 6 months.
In high school students are expected to identify theoretical and practical domains for a variety of functions. In particular, students in secondary school will focus on the special features of quadratic, exponential, and reciprocals of linear functions. These special features provide insights into the real-world situation being modeled. The vertex of a quadratic function could be used to find the maximum or minimum value of the dependent variable while the asymptote of an exponential function could be used to describe the long-term behavior of a situation. Students in high school need to make connections among equations, graphs, tables, and real-world situations of a variety of functions at the high school level. Each representation provides insights into how student make sense of these functions.
Once students are able to identify important features of a variety of functions then they move to seeing how functions can be transformed. Initially students see graphs of functions as a collection of individual points and then move to seeing the graph as an object itself made up of an infinite number of points. Students initially graph functions like f(x) = (x + 3)^{2} and g(x) = x + 2 by calculating points to put in a table, then using these points to create a graph. The more points they create the better the graph looks. They then learn to identify the intercepts and, if applicable, the vertex. This process reinforces the notion of a graph as a collection of a finite set of points. Eventually students are asked to combine functions to create new functions (i.e. find h(x) = f(x) + g(x) or k(x) = g(f(x))). These operations and compositions of functions force students to think of the functions themselves as objects. Students need to view the graphs of functions as objects when they are asked to explain how the graph of a function f(x) compares with the graph of the function m(x) where m(x) = f(x) + 5 or the graph of p(x) where p(x) = f(x + 7). Students at the high school level need to see that a vertical translation is the sum of the original function, and a constant function and a horizontal translation is a composition of the original function with a linear function that has a slope of one; i.e., the graph of the function w(x - 11) is a horizontal translation of the graph of w(x) 11 units to the right.
In high school students gain experience distinguishing among different types of functions and sort them into families. Students learn to identify key features of each type using graphs, tables, equations, and real-world situations. The students then begin to perform operations using the functions themselves as inputs. The results of these operations allow students to create new types of functions that can be used to solve new problems.
All Standard Benchmarks
9.2.1.1
Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain.
9.2.1.2
Distinguish between functions and other relations defined symbolically, graphically or in tabular form.
9.2.1.3
Find the domain of a function defined symbolically, graphically or in a real-world context.
9.2.1.4
Obtain information and draw conclusions from graphs of functions and other relations.
9.2.1.5
Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f(x) = ax^{2} + bx + c, in the form f(x) = a( x - h)^{2} + k , or in factored form.
9.2.1.6
Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function.
9.2.1.7
Understand the concept of an asymptote and identify asymptotes for exponential functions and reciprocals of linear functions, using symbolic and graphical methods.
9.2.1.8
Make qualitative statements about the rate of change of a function, based on its graph or table of values.
9.2.1.9
Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations.
Standard 9.2.1 Benchmark Group B - Parabolas and Asymptotes
9.2.1.5
Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f(x) = ax^{2} + bx + c, in the form f(x) = a( x - h)^{2} + k , or in factored form.
9.2.1.6
Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function.
9.2.1.7
Understand the concept of an asymptote and identify asymptotes for exponential functions and reciprocals of linear functions, using symbolic and graphical methods.
What students should know and be able to do [at a mastery level] related to these benchmarks
- Students should be able to identify that equations like h(t) = -16t^{2} + 20t - 2, f(x) = 3(x - 1)^{2} + 4, and y = 3(x - 3)(x + 1) all represent quadratic functions which are parabolas when graphed on a coordinate plane. They should be able to graph these functions by hand using a table as well as using an electronic graphing tool such as a graphing calculator.
- Students should be able to identify the vertex of a parabola by inspection if the function is given in the vertex form y = a(x - h)2 + k. They should be able to identify the line of symmetry as x = h by knowing that it is a vertical line passing through the vertex. Students should know to use algebraic methods (factoring, completing the square, or quadratic formula) to find the x- and y-intercepts when given the function in this form.
- Students should be able to identify the y-intercept by inspection when given the function in standard form f(x) = ax^{2} + bx + c as (0,c). Students should know to use algebraic methods to find the x-intercepts if they exist. They should be able to find the coordinates of the vertex by completing the square or by knowing that the vertex is at [math](-\frac{b}{2a},f(-\frac{b}{2a}))[/math].
- Students should be able to identify the x-intercepts of a quadratic function given in factored form y = a(x - b)(x - c) as (b,0) and (c,0) by inspection. They should know that because of symmetry, the x-coordinate of the vertex is the midpoint of the x-coordinates of the two x-intercepts.
- Students should understand that the zeroes of a function (also known as the root or solution) can be found at the x-intercept(s) of the function when graphed, at the value of f(x) when x = 0.
- Students should understand how to locate the maximum and minimum values of a function, and describe them by giving the values of the independent variable x and the dependent variable y at these points on the graph.
- Students should be able to identify both vertical and horizontal asymptotes in graphs of exponential functions and the reciprocals of linear functions. They should understand why an asymptote exists and should be able to predict the presence of an asymptote given the symbolic form of these equations.
- Students should be able to graph vertical lines given in the form x = 3. They should be able to give several ordered pairs that are on this line (e.g., (3,0),(3,-2),(3,2.5)) and plot those points on a coordinate plane. They should also be able to explain that the slope of a vertical line is undefined.
- Students should be able to identify the x- and y-intercepts given the graph of a non-linear function. They should be able to identify where on the graph the value of the dependent variable is being maximized or minimized within the domain of the function or within a given interval.
- Students should know how to find the value of dependent variables for very large and very small values of the independent variables of exponential functions (y = a • b^{x}). Given [math]y=3\cdot(\frac{1}{2})^{x}[/math], students should be able to reason that as the value of [math]x[/math] gets larger and larger that the value of [math]y[/math] gets closer and closer to 0. They should be able to reason that the x-axis is the horizontal asymptote and that these asymptotes guide long-term behavior of exponential functions.
- Reciprocals of linear functions are rational functions of the form [math]f(x) = \frac{1}{x+5}[/math]. Students should be able to identify the value of the domain that makes the function undefined. They should further be able to identify both the horizontal asymptote (i.e., the x-axis) and the vertical asymptote (i.e., x = -5). They should be able to explain how the horizontal asymptote guides the function for large and small values of x and that the vertical asymptote guides the graph when the values of x are near 5.
Work from previous grades that supports this new learning includes:
- Represent proportional relationships with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another.
- Represent relationships in various contexts with equations involving variables and positive and negative rational numbers. Use the properties of equality to solve for the value of a variable. Interpret the solution in the original context.
- Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another.
- Identify graphical properties of linear functions including slopes and intercepts. Know that the slope equals the rate of change, and that the y-intercept is zero when the function represents a proportional relationship.
- Identify how coefficient changes in the equation f(x) = mx + b affect the graphs of linear functions. Know how to use graphing technology to examine these effects.
- Represent arithmetic sequences using equations, tables, graphs and verbal descriptions, and use them to solve problems.
- Represent geometric sequences using equations, tables, graphs and verbal descriptions, and use them to solve problems.
- Students entering grades 9 to 11 know how to graph by hand linear functions given in standard form (ax + by = c) or slope-intercept form (y = mx + b) and how to use an electronic graphing tool. They should be able to identify the x-intercept and y-intercept of a linear function using an equation, a table, and a graph.
- Students entering grades 9 to 11 should be able to distinguish between functions that are linear and functions that are not using tables, graphs, equations, and real-world situations. (repeated in next set of benchmarks)
- Students entering grades 9 to 11 should be able to explain whether a slope of a linear function is positive, negative, or zero. (repeated in next set of benchmarks)
- Students entering grades 9 to 11 should be able to calculate the value of the expression a • b^{x} when given the values of a, b, and x (all whole numbers).
- Students entering grades 9 to 11 should be able to multiply fractions fluently and recognize that if one of the two factors in a multiplication problem is between 0 and 1 that the resulting product is less than the other factor.
NCTM Standards: (those that apply to 9.2.1 are bolded)
(p.296, PSSM) Instructional programs from Pre-K through grade 12 should enable all students to:
1. Understand patterns, relations, and functions. In grades 9-12 all students should
- generalize patterns using explicitly defined and recursively defined functions;
- understand relations and functions and select, convert flexibly among, and use various representations for them;
- analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior;
- understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions;
- understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions;
- interpret representations of functions of two variables.
2. Represent and analyze mathematical situations and structures using algebraic symbols. In grades 9-12 all students should
- understand the meaning of equivalent forms of expressions, equations, inequalities, and relations;
- write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency - mentally or with paper and pencil in simple cases and using technology in all cases;
- use symbolic algebra to represent and explain mathematical relationships;
- use a variety of symbolic representations, including recursive and parametric equations, for functions and relations;
- judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.
3. Use mathematical models to represent and understand quantitative relationships. In grades 9-12 all students should
- identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships;
- use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts;
- draw reasonable conclusions about a situation being modeled.
4. Analyze change in various contexts. In grades 9-12 all students should
- approximate and interpret rates of change from graphical and numerical data.
Common Core State Standards (CCSM)
High School: Functions
F-IF.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
F-IF.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F-IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.^{★}
F-IF.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.^{★}
F-IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.^{★}
F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.^{★}
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
b. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
c. Graph exponential functions, showing intercepts and end behavior.
F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as [math]y=(1.02)^{t}[/math], [math]y=(0.97)^{t}[/math], [math]y=(1.01)^{12t}[/math], [math]y=(1.2)^{\frac{t}{10}}[/math], and classify them as representing exponential growth or decay.
F-BF.1. Write a function that describes a relationship between two quantities.^{★}
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
F-LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
F-LE.2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
F-LE.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
F-LE.5. Interpret the parameters in a linear or exponential function in terms of a context.
Misconceptions
Student Misconceptions and Common Errors
- Students interchange the x-intercept and the y-intercept
- Students misunderstand the zero of a function to mean when x = 0, rather than when y = 0.
- The domain of most quadratic functions is the set of all real numbers but many students lack the arithmetic sense to compute with numbers that are not integers. Students may be able to graph the function f(x) = 3x^{2} using integral values for x but would not be able to calculate the exact value of [math]f(\frac{1}{2})[/math].
- Students state that x and y-intercepts are values rather than the coordinates of points on a graph. The y-intercept of the line y = 2x + 7 is the coordinate (0,7) not the value 7. The x-intercepts of the function y = 3(x + 5)(2 - x) are (-5,0) and (2,0) not -5 and 2. This is an important distinction since intercepts highlight important features of the relationship between two variables and not simply the value of one variable.
- Students incorrectly identify the features of a function based on limited information. For example, a student might incorrectly explain that the graph of the function [math]y=\frac{1}{10}x(50-x)[/math] is a linear function because when the function is entered into a graphing calculator using the default settings the graph looks like a line.
- Students will incorrectly state that the maximum value of the function y = -3(x - 11)^{2} + 7 is 11. Students confuse which variable, the independent or dependent, is being maximized or minimized and which variable determines where this occurs.
- Students use end behaviors of a function within a specific graphing window to represent the maximum or minimum values of the function.
- Students may struggle with the concept of asymptotes, including what features of an equation lead to an asymptote in the graph of the function. Students may not realize the effect of the graphing window in viewing asymptotes.
- Students incorrectly state that lines described symbolically as y = 3 (or x = -2) are points rather than a collection of infinitely many ordered pairs that form a horizontal (or vertical) line.
- Students are unable to explain that the slope of a vertical line is undefined.
- Students confuse the meaning of exponents and incorrectly calculate the value of exponential expressions (e.g., 2^{-3} = -8, 2^{0} = 0, or [math]9^{\frac{1}{2}}=4.5[/math]).
- Students incorrectly state that graphs of a quadratic functions and/or exponential functions have vertical asymptotes.
- Students will incorrectly graph rational functions like [math]y=\frac{1}{x}[/math] as shown below where integer values were chosen for [math]x[/math]. The calculated points are just connected.
Vignette
In the Classroom
Most functions can be written in different forms. Each form typically allows different insights into the behavior of the functions. For example, the cubic function f(x) = -3(x + 2)(x - 3)(x - 5) written in factored form makes it easy to identify the zeros and end-term behavior while the standard form of the same function f(x) = -3x^{3} + 18x^{2} + 3x -90 makes it easy to identify the function type and the -intercept. Quadratic functions have many forms, each having features that provide insights into the behavior of the function. The following activity is designed for students who are able to graph quadratics functions by selecting points, making a table, and plotting the points.
Pairs of students are provided with a sheet of equations and are asked to sort the equations in any way that makes sense to them. The intention of this sorting is to allow students to determine what the differences and commonalities among the equations are and decide what is important. The teacher should walk around and ask the students to explain their sorting criteria. The sample equations are shown below.
E1 y = (x + 2)(x - 3) | E2 y = x(x - 3) | E3 y = -x^{2} + 6 |
E4 y = (x - 2)(x + 3) | E5 y = 2(x + 2)(x - 3) | E6 y = (x + 2)(3 - x) |
E7 x = y^{2} | E8 y = 2x^{2} - 5x - 3 | E9 y = (x - 1)(x + 6) |
E10 y = 3(x + 1)^{2} - 2 | E11 y = x^{2} + x4 + 3 | E12 y = -x^{2} - x + 6 |
E13 y = 2(x - 1)^{2} + 3 | E14 y = -(x - 3)^{2} + 6 | E15 y = -2x + 6 |
The sorting criteria developed by the students are then discussed as a whole class. The teacher should try to use the students' vocabulary to come up with classifications.
The students will then be provided with the graphs of these functions and are asked to match the graphs with the equations. The teacher walks around and checks to see how the students are deciding to make the matches. The teacher should try to reinforce the vocabulary y-intercept, x-intercept, and vertex as he or she interacts with students. After students have had some time to make an initial matching they are asked to use a graphing utility to check their answers. The lesson closes with the class discussing the names for each form of a quadratic function and the features that each form highlights in the graph. Sorting activities like this allow students to make and test conjectures about the equations they see rather than memorizing what the teacher tells them. Mathematics classrooms should encourage the making and testing of conjectures a common practice by students and the sorting activity described above illustrates this feature.
Resources
Teacher Notes
- Teachers need to be cognizant of their use of the terms zeroes, solutions, roots, and x-intercepts so that students realize that these terms are interchangeable.
- Teachers need to focus on the understanding of x-intercept and y-intercept to avoid confusion between the two. Students should be able to identify these in tables, graphs, and using the symbolic form of the function.
- Teachers need to show students how the window setting of graphing utilities can affect the features of a function that are visible. Students should know how to choose the appropriate window by examination of the symbolic form of the function.
- Teachers need to ensure that students understand the definitions of minimum and maximum for a function and how to describe that information.
- Instruction on functions of two variables needs to focus on the relationship between an independent variable and a dependent variable. Students first need to see that solutions to equations with two variables like y = x^{2} + 4x have solutions that are ordered pairs. A table of ordered pairs provides a finite number of these solutions, while graphs and equations can show an infinite number of solutions.
- Making connections among the different representations of functions, including tables, graphs, equations, and real-world situations, is key to learning about connections.
- Intercepts provide a visual focus for many types of functions including quadratics. Each form of a quadratic function provides insight to specific features of the graph. Students should be flexible in choosing which form to use in specific situations and know the advantages and disadvantages of each one. Each form highlights an important difference between multiplication and addition. For example, the factors of y = -2(x + 3)(x - 1) highlight the zero-product property
- The concept of an asymptote is very difficult for students because the behavior of a function near an asymptote means working with fractions. Selecting interesting points for the function [math]y=\frac{1}{x}[/math] means dealing with compound fractions like [math]\frac{1}{\frac{1}{5}}[/math]. Teachers need to help students make sense of this fraction as well as making connections to where the resulting ordered-pair [math](\frac{1}{5},5)[/math] appears on the graph. Calculating with fractions is also important to making sense of the horizontal asymptote when working with exponential functions. Students need to be flexible in selecting points that they investigate when working with different functions and this flexibility is enhanced when students are comfortable using a wide range of numbers.
- Using real-world situations to make sense of asymptotes is very important. The half-life of cesium-137 is approximately 30 years and the amount of radioactive material over time can be modeled using an exponential function. The amount of radioactive material approaches the horizontal asymptote of zero over many years just as the graph of an exponential decay function does. Even though we do not want students to have first-hand exposure to cesium-137, we do want them to be able to connect situations that occur in the world to their daily life and their life in mathematics class.
Instructional Resources
- Hanging Chains
This lesson allows students to make connections between points on a graph and the equation of a parabola. - Light it Up
This lesson allows students to make connections among the table, graph, and equation of rational functions using a mirror and flashlight. - Do I Have to Mow the Whole Thing?
This lesson helps student understand the characteristics of inverse variation using tables, graphs, and equations.
Additional Instructional Resources
- 9.2.1.6 Intercepts, Zeros, Roots, Factors, & Solutions
During the activity, students will use the CALCULATE functions, POLYSMLT, and the graphing capabilities of their grapher to analyze the relationship between x-intercepts, zeros, roots, factors, and solutions of polynomial functions/equations.
- 9.2.1.5 Forms of Quadratic Equations (uses TI nspire
This activity is designed as a 1 period summary activity on quadratic equations written in vertex form and intercept form (with one problem expressed in standard form to be solved with the quadratic formula). Students should already be familiar with the forms of the equation as well as basic graphing. Keystrokes are listed in the student worksheet and are relatively simple. Questions appear in both the handheld file and on the worksheet for students to answer in either location.
- 9.2.1.5 Given the Graph of a Parabola, State its Equation in Vertex Form
This activity is designed for students to study on their own. It is designed in a 'StudyCard' format. A graph of a parabola is shown. The student is asked to find the equation of the parabola in vertex form: y = a*(x - h)^(2) + v. Press enter on the double up arrow in the Answer section to see the answer. This does not 'keep score'; it is just a collection of electronic flash cards
- Asymptotes & Zeros
Students relate the graph of a rational function to the graphs of the polynomial functions of its numerator and denominator. Students graph these polynomials one at a time and identify their y-intercepts and zeros. Using the handheld's manual manipulation functions, students can manipulate the graphs of the numerator and denominator functions and see the effect on the rational function.
- Zeroes
- Maximum
- Minimum
- Vertex
- Asymptote
Professional Learning Communities
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
- Am I allowing time for my students to make and test conjectures before telling them the answer?
- Are students able to independently use a graphing utility to graph functions?
- Are my students able to convert easily between forms of a quadratic function?
Materials - suggested articles and books
Developing Essential Understanding of Functions Gr. 9-12, (NCTM, 2010)
Focus in HS Mathematics: Reasoning and Sense Making in Algebra, (NCTM, 2010)
Type II Activities
Bush, W.S. (Ed.). (2000). Mathematics assessment: Cases and discussion questions for grades 6-12.
Bush, W. S., & Greer, A. S. (Eds.). (1999). Mathematics assessment: A practical handbook for grades 9-12.
National Council of Teachers of Mathematics. (2010). Developing essential understandings of functions - Grades 9-12. Reston, VA: NCTM.
See Why the Graph Breaks in NCTM's Mathematics Assessment: Cases and Discussion Questions for Grades 6-12 (p. 24-27; teacher notes on p. 100-102).
See The Golf Shot in NCTM's Mathematics Assessment: A Practical Handbook for Grades 9-12 (p. 123).
Common Core State Standards (http://www.corestandards.org/the-standards/mathematics)
Ruddell, M.R., & Shearer, B.A. (2002). Middle school at-risk students become avid word learners with the vocabulary self-collection strategy (VSS). Journal of Adolescent & Adult Literacy, 45, 352-363.
Assessment
Example Item 1 Source: Maryland High School Assessment Algebra/Data Analysis - Item 15 Correct Answer: C Cognitive Level: Knowledge |
Example Item 2 Source: MCAII Item Sampler 11^{th} Grade Mathematics Correct Answer: D Cognitive Level: Understanding |
Example Item 3 Source: Source: Maryland High School Assessment Algebra/Data Analysis - Item 3 Correct Answer: D Cognitive Level: Knowledge |
Differentiation
- Using technology can help students with special needs to better understand the concepts used in this section. Graphing utilities can help students to connect these concepts in the various representational forms (tables, graphs, and equations).
English Language Learners
- EL will struggle with the various terms for zeroes used interchangeably - roots, x-intercepts, solutions. Using a graphic organizer or a Frayer model to help clarify these terms for students should help.
- Asymptotes is difficult to spell and pronounce; teachers will need to enunciate this term clearly and have ELL students practice frequently. This is a term that is not used in other contexts in English language, and students will likely not be at all familiar with it.
- Use as many non-linguistic approaches as possible - demonstrations, manipulatives, pictures, etc. Have students create their own math dictionaries including definitions they create, pictures, and examples of the terms. Promote the use of the Frayer model, Verbal/visual maps, graphic organizers, and foldables to make sense of new vocabulary. Having a word wall in the math classroom to refer to can be helpful.
Resources for teachers of English Language Learners in Math:
Echevarria, J. J., Vogt, M. E., & Short, D. J. (2009). The SIOP model for teaching mathematics to English learners. Columbus, OH: Allyn & Bacon.
Kersaint, G., Thompson, D. R., & Petkova, M. (2009). Teaching mathematics to English language learners. New York: Routledge.
Texas State University System. (2004). Mathematics for English language learners. Retrieved from http://www.tsusmell.org/ .
This website includes online teaching videos showing best practices along with a classroom lesson bank.
- Paper folding is a great way for students to experience the curve of a parabola.
Folding a Parabola Begin by taking a piece of paper. Draw a line near the bottom and place a point in the middle of the paper as shown below. Fold the paper so that the line covers the point then make a crease. Unfold. Repeat this procedure over 20 times and the folds should look like the picture below. Students should be able to identify the parabola that is formed by the repeated folding. They should also be able to identify where the vertex and the axis of symmetry of the parabola are. The teacher can further the students' study of quadratic functions by showing the students a definition of a parabola: Definition of a parabola (Wikipedia): Given a point (the focus) and a corresponding line (the directrix) on the plane, the locus of points in that plane that are equidistant from them is a parabola. The extension for this activity is for students to find the equation of a parabola given that the focus is at point (0,3) and the directrix is at y = 1 using the definition given above. One way to find the quadratic function is to describe the points on the parabola generally as (x,y) and use the distance formula. The distance between the point (x,y) and the focus is equal to the distance between (x,y) and the directrix. Eventually students should be able to determine algebraically that the equation for the parabola is [math]y=\frac{1}{4}x^{2}+2[/math]. |
Parents/Admin
Administrative/Peer Classroom Observation
Students are: | Teachers are: |
making connections between the various forms of a quadratic function and the graph. | giving students time to discuss the advantages and disadvantages of each form of a quadratic function. |
carrying out experiments that approximate quadratic, exponential functions, and reciprocals of linear functions. | allowing students to use materials to carry out experiments to study the relationship between independent and dependent variables. |
This web site provides java applets that allow students to investigate mathematical ideas in dynamic ways.
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